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«WISB^I 


NON-EUCLIDEAN 
GEOMETRY 

A  CRITICAL  AND 
HISTORICAL   STUDY    OF  ITS    DEVELOPMENT 

BY 
ROBERTO  BONOLA 

Professor  in  the  University  of  Pavia 

AUTHORISED  ENGLISH  TRANSLATION  WITH 
ADDITIONAL  APPENDICES 

BY 

H.  S.  CARSLAW 

Professor  in  the  University  of  Sydney,  N.  S.W. 
WITH  AN  INTRODUCTION 

BY 

FEDERIGO  ENRIQUES 

Professor  in  the  University  of  Bologna 
-«m 


CHICAGO 

THE  OPEN   COURT   PUBLISHING  COMPANY 

1912 


COPYRIGHT    BY 

THE   OPEN    COURT   PUBLISHING    COMPANY 

CHICAGO,    U.  S.  A. 

1912 

All  rights  resewed 


Printed  by  W.  Drugulin,  Leipzig,  (Germany) 


Introduction. 

The  translator  of  this  little  volume  has  done  me -the 
honour  to  ask  me  to  write  a  few  lines  of  introduction.  And 
I  do  this  willingly,  not  only  that  I  may  render  homage  to  the 
memory  of  a  friend^  prematurely  torn  from  life  and  from 
science,  but  also  because  I  am  convinced  that  the  work  of 
Roberto  Bonola  deserves  all  the  interest  of  the  studious. 
In  it,  in  fact,  the  young  mathematician  will  find  not  only 
a  clear  exposition  of  the  principles  of  a  theory  now  classical, 
but  also  a  critical  account  of  the  developments  which 
led  to  the  foundation  of  the  theory  in  question. 

It  seems  to  me  that  this  account,  although  concerned 
with  a  particular  field  only,  might  well  serve  as  a  model 
for  a  history  of  science,  in  respect  of  its  accuracy  and 
its  breadth  of  information,  and,  above  all,  the  sound  philo- 
sophic spirit  that  permeates  it.  The  various  attempts  of 
successive  writers  are  all  duly  rated  according  to  their 
relative  importance,  and  are  presented  in  such  a  way 
as  to  bring  out  the  continuity  of  the  progress  of  science, 
and  the  mode  in  which  the  human  mind  is  led  through 
the  tangle  of  partial  error  to  a  broader  and  broader  view 
of  truth.  This  progress  does  not  consist  only  in  the  ac- 
quisition of  fresh  knowledge,  the  prominent  place  is  taken 
by  the  clearing  up  of  ideas  which  it  has  involved;  and  it 
is  remarkable  with  what  skill  the  author  of  this  treatise  has 
elucidated  the  obscure  concepts  which  have  at  particular 
periods  of  time  presented  themselves  to  the  eyes  of  the 
investigator  as  obstacles,  or  causes  of  confusion.  I  will 
cite  as  an  example  his  lucid  analysis  of  the  idea  of  there 


[V  Introduction. 

being  in  the  case  of  Non-Euclidean  Geometry,  in  contrast 
to  Euclidean  Geometry,  an  absolute  or  natural  measure  of 
geometrical  magnitude. 

The  admirable  simplicity  of  the  author's  treatment, 
the  elementary  character  of  the  constructions  he  employs, 
the  sense  of  harmony  which  dominates  every  part  of  this 
little  work,  are  in  accordance,  not  only  with  the  artistic 
temperament  and  broad  education  of  the  author,  but  also 
with  the  lasting  devotion  which  he  bestowed  on  the  Theory 
of  Non-Euclidean  Geometry  from  the  very  beginning  of 
his  scientific  career.  May  his  devotion  stimulate  others  to 
pursue  with  ideals  equally  lofty  the  path  of  historical  and 
philosophical  criticism  of  the  principles  of  science!  Such 
efforts  may  be  regarded  as  the  most  fitting  introduction 
to  the  study  of  the  high  problems  of  philosophy  in  general, 
and  subsequently  of  the  theory  of  the  understanding,  in 
the  most  genuine  and  profound  signification  of  the  term, 
following  the  great  tradition  which  was  interrupted  by  the 
romantic  movement  of  the  nineteenth  century. 

Bologna,  October   ist^   191 1. 

Federigo  Enriques. 


Translator's  Preface. 

Bonola's  Non-Euclidean  Geometry  is  an  elementary 
historical  and  critical  study  of  the  development  of  that  subject. 
Based  upon  his  article  in  Enriques'  collection  of  Monographs 
on  Questions  of  Elementary  Geometry^,  in  its  final  form  it  still 
retains  its  elementary  character,  and  only  in  the  last  chapter 
is  a  knowledge  of  more  advanced  mathematics  required. 

Recent  changes  in  the  teaching  of  Elementary  Geometry 
in  England  and  America  have  made  it  more  then  ever  ne- 
cessary that  those  who  are  engaged  in  the  training  of  the 
teachers  should  be  able  to  tell  them  something  of  the 
growth  of  that  science;  of  the  hypothesis  on  which  it 
is  built;  more  especially  of  that  hypotheses  on  which  rests 
Euclid's  theory  of  parallels;  of  the  long  discussion  to  which 
that  theory  was  subjected;  and  of  the  final  discovery  of  the 
logical  possibility  of  the  different  Non-Euclidean  Geometries. 

These  questions,  and  others  associated  with  them,  are 
treated  in  an  elementary  way  in  the  pages  of  this  book. 

In  the  English  translation,  which  Professor  Bonola 
kindly  permitted  me  to  undertake,  I  have  introduced  some 
changes  made  in  the  German  translation.^  For  permission 
to  do  so  I  desire  to  express  my  sincere  thanks  to  the  firm  of 
B.  G.  Teubner  and  to  Professor  Liebmann.  Considerable 
new  material  has  also  been  placed  in  my  hands  by  Professor 
Bonola,  including  a  slightly  altered  discussion  of  part  of 


1  Enriques,  F.,  Questioni  riguardan/i  la  geometria  elementare, 
(Bologna,  Zanichelli,   1900). 

2  Wissenschaft  und  Hypothese,  IV.  Band  :  Die  nichteuklidische 
Geometrie.  Historisch-kritische  Darstellung  ihrer  Entwicklung.  Von 
R,  Bonola,     Deutsch  v.  H.  Liebmann.     (Teubner,  Leipzig,   190SJ. 


VI  Translator's  Preface. 

Saccheri's  work,  an  Appendix  on  the  Independence  of  Pro- 
jective Geometry  from  the  Parallel  Postulate,  and  some  further 
Non-Euclidean  Parallel  Constructions. 

In  dealing  with  Gauss's  contribution  to  Non-Euclidean 
Geometry  I  have  made  some  changes  in  the  original  on  the 
authority  of  the  most  recent  discoveries  among  Gauss's 
papers.  A  reference  to  Thibaut's  'proof,  and  some  addit- 
ional footnotes  have  been  inserted.  Those  for  which  I  am 
responsible  have  been  placed  within  square  brackets.  I  have 
also  added  another  Appendix,  containing  an  elementary 
proof  of  the  impossibility  of  proving  the  Parallel  Postulate, 
based  upon  the  properties  of  a  system  of  circles  orthogonal 
to  a  fixed  circle.  This  method  offers  fewer  difficulties  than 
the  others,  and  the  discussion  also  establishes  some  of  the 
striking  theorems  of  the  hyperbolic  Geometry. 

It  only  remains  for  me  to  thank  Professor  Gibson  of 
Glasgow  for  some  valuable  suggestions,  to  acknowledge  the 
interest,  which  both  the  author  and  Professor  Liebmann  have 
taken  in  the  progress  of  the  translation,  and  to  express  my 
satisfaction  that  it  finds  a  place  in  the  same  collection  as 
Hilbert's  classical  Grundlagen  der  Geometrie. 

P.  S.  As  the  book  is  passing  through  the  press  I  have 
received  the  sad  news  of  the  death  of  Professor  Bonola. 
With  him  the  Italian  School  of  Mathematics  has  lost  one  Of 
its  most  devoted  workers  on  the  Principles  of  Geometry. 
Professor  Enriques,  his  intimate  friend,  from  whom  I  heard 
of  Bonola's  death,  has  kindly  consented  to  write  a  short 
introduction  to  the  present  volume.  I  have  to  thank  him, 
and  also  Professor  W.  H.  Young,  in  whose  hands,  to  avoid 
delay,  I  am  leaving  the  matter  of  the  translation  of  this 
introduction  and  its  passage  through  the  press. 

The  University,  Sydney,  August  1 9 1 1 . 

H.  S.  Carslaw. 


Author's  Preface. 

The  material  now  available  on  the  origin  and  develop- 
ment of  Non-Euclidean  Geometry,  and  the  interest  felt  in 
the  critical  and  historical  exposition  of  the  principles  of  the 
various  sciences,  have  led  me  to  expand  the  first  part  of  my 
article — Sulla  teoria  delle  parallele  e  sulle  geometrie  iioji- 
euclidee — which  appeared  sÌ5i  years  ago  in  the  Questioni  ri- 
guardanti la  geometria  elemefiiare,  collected  and  arranged 
by  Professor  F.  Enriques. 

That  article,  which  has  been  completely  rewritten  for  the 
German  translation*  of  the  work,  was  chiefly  concerned  with 
the  systematic  part  of  the  subject.  This  book  is  devoted,  on 
the  other  hand,  to  a  fuller  treatment  of  the  history  of  parallels, 
and  to  the  historical  development  of  the  geometries  of  Lo- 
fi atschewky-Boly  ai  and  RiEMANN. 

In  Chapter  I.,  which  goes  back  to  the  work  of  Euclid 
and  the  earliest  commentators  on  the  Fifth  Postulate,  I  have 
given  the  most  important  arguments,  by  means  of  which 
the  Greeks,  the  Arabs  and  the  geometers  of  the  Renaissance 
attempted  to  place  the  theory  of  parallels  on  a  firmer 
foundation.  In  Chapter  II.,  relying  chiefly  upon  the  work  of 
Saccheri,  Lambert  and  Legendre,  I  have  tried  to  throw 
some  light  on  the  transition  from  the  old  to  the  new 
ideas,  which  became  prevalent  in  the  beginning  of  the  19th 
Century.    In  Chapters  III.   and  IV.,  by  the  aid  of  the  in- 


I  Enriques,  F.,  Fragen  der  Elementargeometrie.  I.  Teil:  Die 
Grundlagen  der  Geometrie.  Deutsch  von  H.  Thieme.  {1910.) 
II.  Teil:  Die  geometrischen  Aufgaben,  ihre  Losung  und  Losbarkeit. 
Deutsch  von  H.  Fleischer.     (1907.)     Teubner,  Leipzig. 


YUj  Author's  Preface. 

vestigations  of  GausS;  Schweikart,  TaurinuS;  and  the  con- 
structive work  of  Lobatschewsky  and  Bolyai,  I  have  ex- 
plained the  principles  of  the  first  of  the  geometrical  systems, 
founded  upon  the  denial  of  Euclid's  Fifth  Hypothesis.  In 
Chapter  V.,  I  have  described  synthetically  the  further  deve- 
lopment of  Non-Euclidean  Geometry,  due  to  the  work  of 
RiEMANN  and  Helmholtz  on  the  structure  of  space,  and 
to  Cayley's  projective  interpretation  of  the  metrical  proper- 
ties of  geometry. 

In  the  whole  of  the  book  I  have  endeavoured  to  pre- 
sent, the  various  arguments  in  their  historical  order.  How- 
ever when  such  an  order  would  have  made  it  impossible  to 
treat  the  subject  simply,  I  have  not  hesitated  to  sacrifice  it, 
so  that  I  might  preserve  the  strictly  elementary  character  of 
the  book. 

Among  the  numerous  postulates  equivalent  to  Euclid's 
Fifth  Postulate,  the  most  remarkable  of  which  are  brought 
together  at  the  end  of  Chapter  IV.,  there  is  one  of  a  statical 
nature,  whose  experimental  verification  would  furnish  an 
empirical  foundation  of  the  theory  of  parallels.  In  this  we 
have  an  important  link  between  Geometry  and  Statics 
(Genocchi);  and  as  it  was  impossible  to  find  a  suitable  place 
for  it  in  the  preceding  Chapters,  the  first  of  the  two  Notes'^ 
in  the  Appendix  is  devoted  to  it. 

The  second  Note  refers  to  a  theory  no  less  interesting. 
The  investigations  of  Gauss,  Lobatschewsky  and  Bolyai  on 
the  theory  of  parallels  depend  upon  an  extension  of  one  of 
the  fundamental  conceptions  of  classical  geometry.  But  a 
conception  can  generally  be  extended  in  various  directions. 
In  this  case,  the  ordinary  idea  of  parallelism,  founded  on 
the  hypothesis  of  non-intersecting  straight  Unes,  coplanar  and 

I  In  the  English  translation  these  Notes  are  called  Appendix  I. 
and  Appendix  II. 


Author's  Preface.  IX 

equidistant,  was  extended  by  the  above-mentioned  geometers, 
who  gave  up  Euclid's  Fifth  Postulate  (equidistance),  and 
later,  by  Clifford,  who  abandoned  the  hypothesis  that  the 
lines  should  be  m  the  same  plane. 

No  elementary  treatment  of  Clifford's  parallels  is  avail- 
able, as  they  have  been  studied  first  by  the  projective 
method  (Clifford-Klein)  and  later,  by  the  aid  of  Different- 
ial-Geometry (BiANCHi-FuBiNi).  For  this  reason  the  second 
Note  is  chiefly  devoted  to  the  exposition  of  their  simplest 
and  neatest  properties  in  an  elementary  and  synthetical 
manner.  This  Note  concludes  with  a  rapid  sketch  of  Clif- 
ford-Klein's problem,  which  is  allied  historically  to  the 
parallelism  of  Clifford.  In  this  problem  an  attempt  is  made 
to  characterize  the  geometrical  structure  of  space,  by  assum- 
ing as  a  foundation  the  smallest  possible  number  of  postul- 
ates, consistent  with  the  experimental  data,  and  with  the 
principle  of  the  homogeneity  of  space. 

This  is,  briefly,  the  nature  of  the  book.  Before  sub- 
mitting the  little  work  to  the  favourable  judgment  of  its 
readers,  I  wish  most  heartily  to  thank  my  respected  teacher, 
Professor  Federigo  Enriques,  for  the  valuable  advice  with 
which  he  has  assisted  me  in  the  disposition  of  the  material 
and  in  the  critical  part  of  the  work;  Professor  Corrado  Segre, 
for  kindly  placing  at  my  disposal  the  manuscript  of  a  course 
of  lectures  on  Non-Euclidean  geometry,  given  by  him,  three 
years  ago,  in  the  University  of  Turin;  and  my  friend.  Professor 
Giovanni  Vailati,  for  the  valuable  references  which  he  has 
given  me  on  Greek  geometry,  and  for  his  help  in  the  cor- 
rection of  the  proofs. 

Finally  my  grateful  thanks  are  due  to  my  publisher 
Cesare  Zanichelli,  who  has  so  readily  placed  my  book  in 
his  collection  of  scientific  works. 

Pavia,  March,  1906. 

Roberto  Bonola. 


Table  of  Contents. 

Chapter  I.  pages 

The  Attempts  to  prove  Euclid's  Parallel  Postulate. 

S    I — 5.    The    Greek    Geometers    and    the    Parallel 

Postulate      I — 9 

S  6.  The  Arabs  and  the  Parallel  Postulate         ..       ..  9 — 12 
S  7 — 10.    The   Parallel  Postulate    during    the  Renais- 
sance and  the   ly^^  Century 12 — 21 

Chapter  II. 

The  Forerunners  of  Non-Euclidean  Geometry. 

S  II — 17.  Gerolamo  Saccheri  (1667 — 1733)     ..      ..  22—44 

S  18 — 22.  Johann   Heinrich  Lambert  (172S — 1777)  44—51 
S  23 — 26.  The  French   Geometers   towards   the  End 

of  the   l8th  Century           51 — 55 

S  27 — 28.  Adrien  Marie  Legendre  (1752 — 1833)  ..  55— 60 

S  29.  Wolfgang  Bolyai  (1775 — 1856)      60—62 

§  30.  Friedrich  Ludwig  Wachter  (1792  — 1817)     ..  62 — 63 

§  30  (bis)  Bernhard  Friedrich  Thib/vut  (1776—1832)  63 

Chapter  III. 

The  Founders  of  Non-Euclidean  Geometry. 

S  31—34-  Karl  Friedrich  Gauss  (1777— 1855)       ..  64 — 75 

S  35.  Ferdinand  Karl  Schweikart  (1780—1859)  ..  75—77 

S  36 — 38.  Franz  Adolf  Taurinus  (1794—1874)      ..  77 — 83 

Chapter  IV. 

The  Founders  of  Non-Euclidean  Geometry  (Cont). 
S  39—45-  Nicolai  Ivanovitsch  Lobatschewsky 

(1793—1856)        , 84—96 

S  46 — 55.  Johann  Bolyai  (1S02  — 1860)     96—113 

S  56—58.  The  Absolute  Trigonometry        I13— 118 

§  59.  Hypotheses  equivalent  to  Euclid's  Postulate    ..  118—121 

§  60  —  65.  The  Spread  of  Non-Euclidean  Geometry  121 — 128 

Chapter  V. 

The  Later  Development  of  Non-Euclidean  Geometry. 

S  66.  Introduction 129 


Table  of  Contents.  XI 

pages 
Differential  Geometry  and  Non-Euclidean  Geometry. 

S  67—69.  Geometry  upon  a  Surface ..  130 — 139 

§  70 — 76.  Principles  of  Plane  Geometry  on  the  Ideas 

of  RiEMANN         139 — 150 

§  77.  Principles  of  Riemann's  Solid  Geometry..       ..  151 — 152 

g  78.  The  Work  of  Helmholtz  and  the  Investigations 

of  Lie 152 — 154 

Projective  Geometry  and  Non-Euclidean  Geometry. 

S  79 "83.    Subordination    of   Metrical    Geometry    to 

Projective  Geometry 154 — 164 

S   84 — 91.  Representation  of  the  Geometry  of  Lobat- 

SCHEWSKV-BOLYAI  On  the  Euclidean  Plane   ..       ..  164—175 

S  92.  Representation  of  Riemann's  Elliptic  Geometry 

in  Euclidean  Space 175 — 176 

S  93.  Foundation  of  Geometry  upon  Descriptive  Pro- 
perties   ,  176  —  177 

S  94.  The  Impossibility  of  proving  Euclid's  Postulate  177 — ^^o 

Appendix  I. 

The    Fundamental    Principles    of    Statics    and    Euclid's 

Postulate. 

S  I — 3.  On  the  Principle  of  the  Lever        181  — 184 

S  4 — 8.    On    the    Composition    of   Forces    acting    at 

a  Point 184  —  192 

S  9 — 10.  Non-Euclidean  Statics 192  —  195 

S  II  — 12.    Deduction    of   Plane  Trigonometry    from 

Statics 195 — 199 

Appendix  II. 

Clifford's  Parallels  and  Surface.     Sketch  of  Clifford- 
Klein's  Problem. 

S  1—4.  Clifford's  Parallels 2co— 206 

S  5—8.     Clifford's  Surface 206—211 

S  9  —  11.  Sketch  of  Clifford-Klein's  Problem         ..     211  —  215 
Appendix  III. 

The    Non-Euclidean    Parallel    Construction    and    other 
Allied  Constructions. 
S  1 — 3.  The  Non-Euclidean  Parallel  Construction     ..     216—222 
§  4.  Construction    of  the  Common  Perpendicular   to 

two  non-intersecting  Straight  Lines        222-— 223 

S  5.  Construction    of    the    Common   Parallel    to    the 

Straight  Lines  which  bound  an  Angle 223—224 


XII  Table  of  Contents. 

pages 
S  6.  Construction  of  the  Straight  Line  which  is  per- 
pendicular to  one  of  the  lines  bounding  an  acute 

Angle  and  Parallel  to  the  other 224 

S  7-  The  Absolute  and  the  Parallel  Construction      ..     224 — 226 
Appendix  IV. 

The    Independence  of  Projective  Geometry  from  Euclid's 
Postulate. 

S  I.  Statement  of  the  Problem 227 — 228 

§  2.  Improper   Points    and    the    Complete   Projective 

Plane 228—229 

§  3.  The  Complete  Projective  Line      229 

S  4.  Combination  of  Elements      229 — 231 

§  5.  Improper  Lines        231 — 233 

S  6.  Complete  Projective  Space       233 

S  7.  Indirect    Proof    of    the    Independence    of   Pro- 
jective Geometry  from  the  Fifth  Postulate  ..     233 — 234 
S  8.  Beltrami's   Direct  Proof  of  this  Independence     234—236 
S  9.  Klein's  Direct  Proof  of  this  Independence       ..     236— -237 
Appendix  V. 

The  Impossibility  of  proving  Euclid's  Postulate. 
An    Elementary    Demonstration     of     this    Impossibility 
founded    upon    the   Properties    of  the  System   of 
Circles  orthogonal  to  a  Fixed  Circle. 

§  I.  Introduction      238 

S  2 — 7.    The  System    of  Circles    passing    through    a 

Fixed  Point  239     250 

S  8  — 12.    The    System    of    Circles    orthogonal    to    a 

Fixed  Circle        - 250—264 

Index  of  Authors ..  265 


Chapter  I. 

The  Attempts  to  prove  Euclid's  Parallel 
Postulate 

The   Greek   Geometers    and  the  Parallel  Postulate, 

§  I.  Euclid  (circa  330 — 275,  B.  C.)  calls  two  straight 
lines  parallel,  when  they  are  in  the  same  plane  and  being 
produced  indefinitely  in  both  directions,  do  not  meet  one 
another  in  either  direction  (Def.  XXIII.).^  He  proves  that 
two  straight  lines  are  parallel,  when  they  form  with  one  of 
their  transversals  equal  interior  alternate  angles,  or  equal 
corresponding  angles^  or  interior  angles  on  the  same  side 
which  are  supplementary.  To  prove  the  converse  of  these 
propositions  he  makes  use  of  the  following  Postulate  (V.)  : 

If  a  straight  Ime  falling  on  t7V0  straight  lines  make  the 
ifiterior  angles  on  the  same  side  less  than  two  right  angles, 
the  two  straight  lines,  if  produced  indefinitely,  meet  on  that 
side  on  which  are  the  angles  less  than  the  two  right  angles. 

The  Euclidean  Theory  of  Parallels  is  then  completed 
by  the  following  theorems: 

Straight  lines  which  are  parallel  to  the  same  straight 
line  are  parallel  to  each  other  (Bk.  I.,  Prop.  30). 


I  With  regard  to  Euclid's  text,  references  are  made  to  the 
critical  edition  of  J.  L.  Heiberg  (Leipzig,  Teubner,  1883).  [The 
wording  of  this  definition  (XXIIF,  and  of  Postulate  V  below,  are 
taken  from  Heath's  translation  of  Heiberg's  text.  (Camb.  Univ.  Press, 

1908Ì.] 

I 


2  I.     The  Attempts  to  prove  Euclid's  Parallel  Postulate. 

Through  a  given  point  one  and  only  one  straight  line 
can  be  drawn  which  will  be  parallel  to  a  given  straight  line 
(Bk.  I.  Prop.  31). 

The  straight  lines  joining  the  extremities  of  two  equal 
and  parallel  straight  lines  are  equal  and  parallel  (Bk.  I. 
Prop.  33). 

From  the  last  theorem  it  can  be  shown  that  two  parallel 
straight  lines  are  equidistant  from  each  other.  Among  the 
most  noteworthy  consequences  of  the  Euclidean  theory  are 
the  well-known  theorem  on  the  sum  of  the  angles  of  a  tri- 
angle, and  the  properties  of  similar  figures. 

§  2.  Even  the  earliest  commentators  on  Euclid's  text 
held  that  Postulate  V.  was  not  sufficiently  evident  to  be 
accepted  without  proof,  and  they  attempted  to  deduce  it  as 
a  consequence  of  other  propositions.  To  carry  out  their  pur- 
pose, they  frequently  substituted  other  definitions  of  parallels 
for  the  Euclidean  definition,  given  verbally  in  a  negative 
form.  These  alternative  definitions  do  not  appear  in  this 
form,  which  was  believed  to  be  a  defect. 

Proclus  (410 — 485)  —  in  his  Commefitary  on  the  First 
Book  of  Euclid^  —  hands  down  to  us  valuable  informa- 
tion upon  the  first  attempts  made  in  this  direction.  He  states, 
for  example,  that  Posidonius  (i^'  Century,  B.  C.)  had  pro- 
posed to  call  two  equidistant  and  coplanar  straight  lines  par- 
allels. However,  this  definition  and  the  Euclidean  one 
correspond  to  two  facts,  which  can  appear  separately,  and 


»  Wher  the  text  of  Proclus  is  quoted,  we  refer  to  the  edi- 
tion of  G.  FriEDLEIN:  Frodi  Diadochi  in  primum  Eudidis  element- 
orum  librian  commeyitarii,  [Leipzig,  Teubner,  1873).  [Compare  also 
W.  B.  Frankland,  The  First  Book  of  Eudid''s  Elements  with  a 
Commentary  based  prindpally  upon  that  of  Produs  Diadochus,  (Camb. 
Univ.  Press,  1905).  Also  Heath's  Euclid,  Vol.  I.,  Introduction, 
Chapter  IV.,  to  which  most  important  work  reference  has  been 
made  on  p.  l]. 


The  Commentary  of  Proclus.  •» 

Proclus  (p.  177),  referring  to  a  work  by  Geminus  (1^'  Cen- 
tury, B.  C),  brings  forward  in  this  connection  the  examples 
of  the  hyperbola  and  the  conchoid,  and  their  behaviour  with 
respect  to  their  asymptotes,  to  show  that  there  might  be 
parallel  lines  in  the  Euclidean  sense,  (that  is,  lines  which 
produced  indefinitely  do  not  meet),  which  would  not  be 
parallel  in  the  sense  of  Posidonius,  (that  is,  equidistant). 

Such  a  fact  is  regarded  by  Geminus,  quoting  still  from 
Proclus,  as  the  most  paradoxical  [TrapaòoHÓTaTOv]  in  the 
whole  of  Geometry. 

Before  we  can  bring  Euclid's  definition  into  line 
with  that  of  Posidonius,  it  is  necessary  to  prove  that  if  two 
coplanar  straight  lines  do  not  meet,  they  are  equidistant;  or, 
that  the  locus  of  points,  which  are  equidistant  from  a  straight 
line,  is  a  straight  line.  And  for  the  proof  of  this  proposition 
Euclid  requires  his  Parallel  Postulate. 

However  Proclus  (p.  364)  refuses  to  count  it  among 
the  postulates.  In  justification  of  his  opinion  he  remarks 
that  its  converse  {The  sum  of  hvo  angles  of  a  triangle  is  less 
than  two  right  angles),  is  one  of  the  theorems  proved  by 
Euclid. (Bk.  I.  Prop.  17); 
and  he  thinks  it  impossible 
that  a  theorem  whose  con- 
verse can  be  proved,  is  not 
itself  capable  of  proof.  Also 
he  utters  a  warning  against 
mistaken  appeals  to  self- 
evidence,  and  insists  upon 
the  (hypothetical)  possibi- 
lity of  straight  lines  which 
are  asymptotic  (p.  191 — 2). 

Ptolemy  (2°^  Century,  A.  D.) — we  quote  again  from 
Proclus  (p.  362 — 5)— attempted  to  settle  the  question  by 
means  of  the  following  curious  piece  of  reasoning. 

1* 


A  I.     The  Attempts  to  prove  Euclid's  Parallel  Postulate. 

Let  AB,  CD,  be  two  parallel  straight  lines  and  FG  a 
transversal  (Fig.  i). 

Let  a,  P  be  the  two  interior  angles  to  the  left  of  FG, 
and  a,  P'  the  two  interior  angles  to  the  right. 

Then  a  +  P  will  be  either  greater  than,  equal  to,  or  less 
than  a  +  p'. 

It  is  assumed  that  if  any  one  of  these  cases  holds  for 
one  pair  of  parallels  (e.  g.  a  +  P  ^  2  right  angles)  this  case 
will  also  hold  for  every  other  pair. 

Now  FB,  GD,  are  parallels;  as  are  also  FA  and  GC. 

Since  a  +  P  ^  2  right  angles, 

it  follows  that  a'  +  P'  ^  2  right  angles. 

Thus     a+P  +  a'+P'>>4  right  angles, 
which  is  obviously  absurd. 
Hence  a  +  P  cannot  be  greater  than  2  right  angles. 

In  the  same  way  it  can  be  shown  that 

a  +  P  cannot  be  less  than  2  right  angles. 

Therefore  we  must  have 

a  +  p  =  2  right  angles  (Proclus,  p.  365). 
From  this  result  Euclid's  Postulate  can  be  easily  obtained. 

§  3.  Proclus  (p.  371),  after  a  criticism  of  Ptolemy's 
reasoning,  attempts  to  reach  the  same  goal  by  another  path. 
His  demonstration  rests  upon  the  following  proposition, 
which  he  assumes  as  evident: — The  distance  between  two 
points  upon  two  intersecting  straight  lines  can  be  made  as  great 
as  7V e  please,  by  prolonging  the  two  lines  sufficiently} 

From  this  he  deduces  the  lemma  :  A  straight  line  which 
meets  one  of  two  parallels  must  also  meet  the  other. 


I  For  the  truth  of  this  proposition,  which  he  assumes  as  self- 
evident,  Proclus  relies  upon  the  authority  of  Aristotle.  Cf. 
De  Coelo  I.,  5.  A  rigorous  demonstration  of  this  very  theorem 
was  given  by  Saccheri  in  the   work  quoted  on  p.  22. 


Proclus  (continued).  C 

His  proof  of  this  lemma  is  as  follows: 
Let  AB,  CD,  be  two  parallels  and  £G  a  transversal, 
cutting  the  former  in  ^  (Fig.  2). 


—  D 


Fig.  2. 

The  distance  of  a  variable  point  on  the  ray  J^G  from 
the  line  AB  increases  without  limit,  when  the  distance  of  that 
point  from  jF  is  increased  indefinitely.  But  since  the  distance 
between  the  two  parallels  is  finite,  the  straight  line  EG  must 
necessarily  meet  CD. 

Proclus,  however,  introduced  the  hypothesis  that  the 
distance  between  two  parallels  remains  finite;  and  from  this 
hypothesis  Euclid's  Parallel  Postulate  can  be  logically  de- 
duced. 

§  4.  Further  evidence  of  the  discussion  and  research 
among  the  Greeks  regarding  Euclid's  Postulate  is  given  by 
the  following  paradoxical  argument.  Relying  upon  it,  accord- 
ing to  Proclus,  some  held  that  it  had  been  shown  that  two 
straight  lines,  which  are  cut  by  a  third,  do  not  meet  one 
another,  even  when  the  sum  of  the  interior  angles  on  the 
same  side  is  less  than  two  right  angles. 

Let  -(4C  be  a  transversal  of  the  two  straight  lines  AB, 
CD  and  let  E  be  the  middle  point  of  AC  (Fig.  3). 

On  the  side  of  ^C  on  which  the  sum  of  the  two  internal 
angles  is  less  than  two  right  angles,  take  the  segments  AF 
and  CG  upon  AB  and  CD  each  equal  to  AE.  The  two 
lines  AB  and  CD  cannot  meet  between  the  points  AF  and 
CG,  since  in  any  triangle  each  side  is  less  than  the  sum  of 
the  other  t\vo. 


A 

1 

~       K 

E 
C 

H 

( 

Ì        L 

_D 


6  I.     The  Attempts  to  prove  Euclid's  Parallel  Postulate. 

The  points  F  and  G  are  then  joined,  and  the  same 
process  is  repeated,  starting  from  the  hne  FG.  The  segments 
FK  and  GL  are  now  taken  on  AB  and  CD,  each  equal  to 
half  of  FG.  The  two  lines  AB,  CD  are  not  able  to  meet 
between  the  points  7%  K  and  G,  L. 

Since  this  operation 
can  be  repeated  indefini- 
tely, it  is  inferred  that  the 
two  lines  AB,  CD  will  never 
meet. 

The  fallacy  in  this  ar- 
gument is  contained  in  the 
use  of  infinity,  since  the 
segments  AF,  FK  could 
tend  to  zero,  while  their 
sum  might  remain  finite.  The  author  of  this  paradox  has 
made  use  of  the  principle  by  means  of  which  Zeno  (495 — 
435  B.  C.)  maintained  that  it  could  be  proved  that  Achilles 
would  never  overtake  the  tortoise,  though  he  were  to  travel 
with  double  its  velocity. 

This  is  pointed  out,  under  another  form,  by  Proclus 
(p.  369 — 70),  where  he  says  that  this  argument  proves  that 
the  point  of  intersection  of  the  lines  could  not  be  reached 
(to  determine,  ópiZieiv)  by  this  process.  It  does  not  prove 
that  such  a  point  does  not  exist.* 

Proclus  remarks  further  that  'since  the  sum  of  two 
angles  of  a  triangle  is  less  than  two  right  angles  (Euclid  Bk.  I. 
Prop.  17),  there  exist  some  lines,  intersected  by  a  third, 
which  meet   on  that  side  on  which  the  sum  of  the  interior 


Fig-  3- 


I  [Suppose  we  start  with  a  triangle  ABC  and  bisect  the  base 
BC  in  D.  Then  on  BA  take  the  segment  BE  equal  to  BD,  and 
on  CA  the  segment  CB'  equal  to  CD,  and  join  EF.  Then  repeat 
this  process  indefinitely.  The  vertex  A  can  never  be  reached  by 
this  means,  although  it  is  at  a  finite  distance.] 


Proclus  (continued).  j 

angles  is  less  than  two  right  angles.  Thus  if  it  is  asserted 
that  for  every  difference  between  this  sum  and  two  right 
angles  the  lines  do  not  meet,  it  can  be  replied  that  for 
greater  differences  the  lines  intersect.' 

'But  if  there  exists  a  point  of  section,  for  certam  pairs 
of  lines,  forming  with  a  third  interior  angles  on  the  same 
side  whose  sum  is  less  than  two  right  angles,  it  remains  to  be 
shown  that  this  is  the  case  for  all  the  pairs  of  lines.  Since 
it  might  be  urged  that  there  could  be  a  certain  deficiency  (from 
two  right  angles)  for  which  they  (the  lines)  would  not  inter- 
sect, while  on  the  other  hand  all  the  other  lines,  for  which  the 
deficiency  was  greater,  would  intersect.'    (Proclus,  p.  371.) 

From  the  sequel  it  will  appear  that  the  question,  which 
Proclus  here  suggests,  can  be  answered  in  the  affirmative 
only  in  the  case  when  the  segment  AC  of  the  transversal 
remains  unaltered,  while  the  lines  rotate  about  the  points  A 
and  C  and  cause  the  difference  from  two  right  angles  to  vary. 

§  5.  Another  very  old  proof  of  the  Fifth  Postulate, 
reproduced  in  the  Arabian  Commentary  of  Al-Nirizi'  (ptt 
Century),  has  come  down  to  us  through  the  Latin  translation 
of  Gherardo  da  Cremona*  (12th  Century),  and  is  attributed 
to  Aganis.3 

The  part  of  this  commentary  relating  to  the  definitions, 
postulates  and  axioms,  contains  frequent  references  to  the 


1  Cf.  R.  O.  Besthorn  u.  J.  L.  Heiberg,  'Codex  Leidensis,' 
399,  I.  Euclidis  Elementa  ex  interpretatione  Al-Hadsckdschadsch  cum 
commentariis  Al-N^ariziif  (Copenhagen,  F.  Hegel,   1893 — 97)- 

2  Cf.  M.  CuRTZE,  *Anariin  in  decern  libros  priores  eleinentoriim 
Euclidis  Commentarii.'  Ex  interpretatione  Gherardi  Cremonensis  in 
Codice  Cracoviensi  569  servata,  (Leipzig,  Teubner,  1899). 

3  With  regard  to  Aganis  it  is  right  to  mention  that  he  is 
identified  by  Curtze  and  Heiberg  with  Geminus.  On  the  other 
hand  P.  Tannery  does  not  accept  this  identification.  Cf.  Tannery, 
*Z<f  phibsophe  Aganis  est-il  identique  à  Geminus?'  Bibliotheca  Math. 
(3)  Bd.  II.  p.  9— II   [1901], 


3  I.     The  Attempts  to  prove  Euclid's  Parallel  Postulate. 

the  name  of  Sambelichius,  easily  identified  with  Simplicius, 
the  celebrated  commentator  on  Aristotle,  who  lived  in  the 
6^^  Century.  It  would  thus  appear  that  Simplicius  had  written 
an  Introduction  to  the  First  Book  of  Euclid,  in  which  he  ex- 
pressed ideas  similar  to  those  of  Geminus  and  Posidonius, 
affirming  that  the  Fifth  Postulate  is  not  self  evident,  and 
bringing  forward  the  demonstration  of  his  friend  Aganis. 

This  demonstration  is  founded  upon  the  hypothesis  that 
equidistant  straight  lines  exist.  Aganis  calls  these  parallels, 
as  had  already  been  done  by  Posidonius.  From  this  hypo- 
thesis he  deduces  that  the  shortest  distance  between  two 
parallels  is  the  common  perpendicular  to  both  the  lines: 
that  two  straight  lines  perpendicular  to  a  third  are  parallel 
to  each  other:  that  two  parallels,  cut  by  a  third  line,  form 
interior  angles  on  the  same  side,  which  are  supplementary, 
and  conversely. 

These  propositions  can  be  proved  so  easily  that  it  is 
unnecessary  for  us  to  reproduce  the  reasoning  of  Aganis. 
Having  remarked  that  Propositions  30  and  33  of  the  First 
Book  of  Euclid  follow  from  them,  we  proceed  to  show  how 
Aganis  constructs  the  point  of  intersection  of  two  straight 
lines  which  are  not  equidistant. 

Let  AB^  GD  be  two  straight  lines  cut  by  the  trans- 
versal EZ^  and  such  that  the  sum  of  the  interior  angles  AEZ^ 
EZD  is  less  than  two  right  angles  (Fig.  4). 

Without  making  our  figure  any  less  general  we  may  sup- 
pose that  the  angle  AEZ  is  a  right  angle. 

Upon  ZD  take  an  arbitrary  point  T. 

From  T  draw  TL  perpendicular  to  ZE. 

Bisect  the  segment  EZ  at  P:  then  bisect  the  segment 
PZ  at  M:  and  then  bisect  the  segments  MZ,  etc.  .  .  .  until 
one  of  the  middle  points  P,  M, .  .  .  falls  on  the  segment  LZ. 

Let  this  point,  for  example,  be  the  point  M. 

Draw  MN  perpendicular  to  EZ,  meeting  ZD  in  N', 


Equidistant  Straight  Lines.  g 

Finally   from  Z£>  cut  off  the  segment  ZC,  the  same 
multiple  of  ZiV  as  Z£  is  of  ZM. 

In  the  case  taken  in  the  figure  ZC  =  4  ZiV. 

The  point  C  thus  obiaified  is  the  point  of  intersection  of 
the  two  straight  ii?ies  AB  and  GD. 
C  F 


G 


To  prove  this  it  would  be  necessary  to  show  that  the 
equal  segments  ZN,  JVS,  .  .  .,  which  have  been  cut  off  one 
after  the  other  firom  the  line  ZD,  have  equal  projections  on 
Z£.  We  do  not  discuss  this  point,  as  we  must  return  to  it 
later  (p.  11).  In  any  case  the  reasoning  is  suggested  directly 
by  Aganis'  figure. 

The  distinctive  feature  of  the  preceding  construction  is 
to  be  noticed.  It  rests  upon  the  (implicit)  use  of  the  so-called 
Postulate  of  Archimedes,  which  is  necessary  for  the  deter- 
mination of  the  segment  J/Z,  less  than  LZ  and  a  submult- 
iple of  EZ. 

The  Arabs  and  the  Parallel  Postulate. 

§  6.  The  Arabs,  succeeding  the  Greeks  as  leaders  in 
mathematical  discovery,  like  them  also  investigated  the  Fifth 
Postulate. 

Some,  however,  accepted  without  hesitation  the  ideas 
and  demonstrations  of  their  teachers.  Among  this  number  is 
Al-Nirlzi  (9th  Century),  whose  commentary  on  the  definitions, 


IO        I.     The  Attempts  to  prove  Euclid's  Parallel  Postulate. 

postulates  and  axioms  ot  the  First  Book  is  modelled  on  the 
Introduction  to  the  ''Elements^  of  Simplicius,  while  his  demon- 
stration of  the  Fifth  Euclideari  Hypothesis  is  that  of  Aganis,  to 
which  we  have  above  referred. 

Others  brought  their  own  personal  contribution  to  the 
argument.  Nasìr-Eddìn  [1201 — 1274],  for  example,  although 
in  his  proof  of  the  Fifth  Postulate  he  employs  the  criterion 
used  by  Aganis,  deserves  to  be  mentioned  for  his  original  idea 
of  explicitly  putting  in  the  forefront  the  theorem  on  the  sum 
of  the  angles  of  a  triangle,  and  for  the  exhaustive  nature  of 
his  reasoning.' 

The  essential  part  of  his  hypothesis  is  as  follows:  If  two 
straight  lines  ;-  afid  s  are  the  07ie  perpendictilar  and  the  other 
oblique  to  the  segment  AB,  the  perpendiculars  drawn  frotn  s 
upon  r  are  less  than  AB  on  the  side  on  7vhich  s  makes  an  acute 
angle  with  AB,  and  greater  on  the  side  on  which  s  makes  an 
obtuse  angle  with  AB. 

It  follows  immediately  that  \i  AB  and  A'B'  are  two  equal 
perpendiculars  to  the  line  BB'  from  the  same  side,  the  line 
AA'  is  itself  perpendicular  to  both  AB  and  A'B'.  Further 
we  have  AA'  =  BE'  ;  and  therefore  the  figure  AA'B'B  is  a 
quadrilateral  with  its  angles  right  angles  and  its  opposite  sides 
equal,  i.  e.,  a  rectangle. 

From  this  result  Nasìr-Eddìn  easily  deduced  that  the  sum 
of  the  angles  of  a  triangle  is  equal  to  two  right  angles.  For 
the  right-angled  triangle  the  theorem  is  obvious,  as  it  is  half 
of  a  rectangle;  for  any  triangle  we  obtain  it  by  breaking  up 
the  triangle  into  two  right-angled  triangles. 

With  this  introduction,  we  can  now  explain  shortly  how 
the  Arabian  geometer  proves  the  Euclidean  Postulate  [cf. 
Aganis]. 

I  Cf.  :  Eiiclidis  elementorum  libri  XII  studii  N'assiredini,  (Rome, 
1594).  This  work,  written  in  Arabic,  was  republished  in  1657  and 
1801.     It  has  not  been  translated  into  any  other  language. 


Nasìr-Eddìn's  Proof. 


II 


o'c  m'  k'  h'  a 

Fig-  5- 


Let  AB,  CD  be  two  rays,  the  one  oblique  and  the  other 
perpendicular  to  the  straight  Hne  AC  (Fig.  5).  From  AB  cut 
oflf  the  part  AH,  and  from  ZTdraw  the  perpendicular  HH' 
to  AC.  If  the  point  H'  falls  on  C,  or  on  the  opposite  side 
of  C  from  A^  the  two  rays  AB  and 
CD  must  intersect.  If,  however,  H' 
falls  between  A  and  C,  draw  the  line 
AL  perpendicular  to  AC  and  equal 
to  HH' .  Then,  from  what  we  have 
said  above,  HL  =  AH' .  In  AH^io- 
<iuced  take  HK  equal  to  AH.  From 
K  draw  KK'  perpendicular  to  AC. 
Since  A'A"  ^  HH',  we  can  take 
X'L'  =  i^'^,  and  join  L'H  The 
quadrilaterals  K'H'HL',  H'ALHaxQ  both  rectangles.  There- 
fore the  three  points  Z',  ZT,  Z  are  in  one  straight  line.  It  fol- 
lows that  ^L'HK=  <^AHL,  and  that  the  triangles  AHL, 
HL'K  diXQ:  equal.  Thus  L'H=  HL,  and  from  the  properties 
of  rectangles,  K'H'  =  H'A. 

In  HK  produced,  take  KM  equal  to  KH.  From  M 
draw  MM'  perpendicular  to  AC.  By  reasoning  similar  to 
•what  has  just  been  given,  it  follows  that 

M'K'  =  K'H  =  H'A. 

This  result  obtained,  we  take  a  multiple  of  AH'  greater 
than  AC  [The  Postulate  of  Archimedes].  For  example,  let 
AO.,  equal  to  4  AH' ,  be  greater  than  AC.  Then  from  AB 
cuttoff  AO  =  4  AH,  and  draw  the  perpendicular  from  O 
to  AC. 

This  perpendicular  will  evidently  be  00' .  Then,  in  the 
right-angled  triangle  AO' O,  the  line  CD,  which  is  perpendicu- 
lar to  the  side  AO' ,  cannot  meet  the  other  side  00' ,  and  it 
must  therefore  meet  the  hypotenuse  OA. 

By  this  means  it  has  been  proved  that  two  straight  lines 
AB,  CD,  must  intersect,  when  one  is  perpendicular  to  the 


12        I-     The  Attempts  to  prove  Euclid's  Parallel  Postulate. 

transversal  AC  and  the  other  obhque  to  it.  In  other  words 
the  Euclidean  Postulate  has  been  proved  for  the  case  in  which 
one  of  the  internal  angles  is  a  right  angle. 

Nasìr-Eddìn  now  makes  use  of  the  theorem  on  the  sum 
of  the  angles  of  a  triangle,  and  by  its  means  reduces  the 
general  case  to  this  particular  one.  We  do  not  give  his  reas- 
oning, as  we  shall  have  to  describe  what  is  equivalent  to 
it  in  a  later  article,    [cf  p.  37.]^ 


The  Parallel  Postulate  during  the  Renaissance  and 
the  i7^b  Century. 

§  7.  The  first  versions  of  the  Elements  made  in  the 
12th  and  13th  Centuries  on  the  Arabian  texts,  and  the  later 
ones,  made  at  the  end  of  the  1 5th  and  the  beginning  of  the 
i6tl»,  based  on  the  Greek  texts,  contain  hardly  any  critical 
notes  on  the  Fifth  Postulate.  Such  criticism  appears  after  the 
year  1550,  chiefly  under  the  influence  of  the  Commentary  of 
Froclus.'^  To  follow  this  more  easily  we  give  a  short  sketch 
of  the  views  taken  by  the  most  noteworthy  commentators  of 
the  1 6th  and  17  th  centuries. 

F.  CoMM ANDINO  [1509 — 1575]  adds  to  the  Euclidean 
definition  of  parallels,  without  giving  any  justification  for  this 


1  Nasìr-Eddìn's  demonstration  of  the  Fifth  Postulate  is  given 
in  full  by  the  English  Geometer  J.  Wallis,  in  Vol.  II.  of  his  works 
(cf.  Note  on  p.  15),  and  by  G.  Castillon,  in  a  paper  published  in 
the  Mém.  de  I'Acad.  roy.  de  Sciences  et  Belles-Lettres  of  Berlin, 
T.  XVIII.  p.  175—183,  (1788— 17S9).  In  addition,  several  other 
writers  refer  to  it,  among  whom  we  would  mention  chiefly,  G.  S. 
Klijgel,  (cf.  note,  (3),  p.  44),  J.  Hoffman,  Kriiik  der  Parallelentheorie^ 
(Jena,  1807);  V.  Flauti,  Niiova  ditnostr azione  del  postulato  quinto,  (Na- 
ples, 1818). 

2  The  Cotnmentary  of  Proclus  was  first  printed  at  Basle  (1533) 
in  the  original  text;  and  next  at  Padua  (1560)  in  Barozzi's  Latin 
translation. 


Italian  Mathematicians  of  the  Renaissance. 


13 


step,  the  idea  of  equidistance.  With  regard  to  the  Fifth  Postul- 
ate he  gives  the  views  and  the  demonstration  of  Proclus/ 

C.  S.  Clavio  [1537 — 1612],  in  his  Latin  translation  of 
Eudid's  text^,  reproduces  and  criticises  the  demonstration  of 
Proclus.  Then  he  brings  forward  a  new  demonstration  of  the 
Euclidean  hypothesis,  based  on  the  theorem:  The  line  equi- 
distant from  a  straight  line  is  a  straight  line;  which  he  at- 
tempts to  justify  by  similar  reasoning.  His  demonstration 
has  many  points  in  common  with  that  of  Nasir-Eddin. 

P.  A.  Cataldi  [? — 1626]  is  the  first  modern  mathema- 
tician to  pubhsh  a  work  devoted  exclusively  to  the  theory  of 
parallels. 3  Cataldi  starts  from  the  conception  of  equidistant 
and  non-equidistant  straight  lines;  but  to  prove  the  effective 
existence  of  equidistant  straight  lines,  he  adopts  the  hypothesis 
that  straight  lines  which  are  not  equidistant  converge  in  one 
direction  and  diverge  in  the  other,  [cf  Nasìr-Eddìn.]  ♦. 

G.  A.  BoRELLi  [1608 — 1679]  takes  the  following  Axiom 
[XIV],  and  attempts  to  justify  his  assumption: 

^If  a  straight  line  which  remains  always  in  the  saine  plane 
as  a  second  straight  line,  moves  so  that  the  o?ie  end  always  touches 
this  line,  and  during  the  whole  displacement  the  first  remains 
continually  perpendicular  to  the  second.,  then  the  other  end,  as  it 
moves,  will  describe  a  straight  line.' 

Then  he  shows  that  two  straight  lines  which  are  perpen- 
dicular to  a  third  are  equidistant,  and  he  defines  parallels  as 
equidistant  straight  lines. 

The  theory  of  parallels  follows.  5 

1  Elementonim  libri  XV,  (Pesaro,  1572). 

2  Euclidis  elementorum  libri  XV,  (Rome,   1574). 

3  OpereUa  delle  linee  rette  equidistanti  et  non  equidistanti,  (Bologna, 
1603). 

4  Cataldi  made  some  further  additions  to  his  argument  in  the 
work,  Aggiunta  all'  operetta  delle  linee  rette  equidistanti  et  noti  equi- 
distanti.    (Bologna,  1604). 

5  BORELLI:  Euclides  restitutus,  (Pisa,  1658). 


I A         I-     The  Attempts  to  prove  Ex:clid's  Parallel  Postulate. 

§  8.  Giordano  Vitale  [1633  — 171 1]  again  returns  to 
the  idea  of  equidistance  put  forward  by  Posidonius,  and  re- 
cognizes, with  Proclus,  that  it  is  necessary  to  exclude  the  pos- 
sibiHty  of  the  Euchdean  parallels  being  asymptotic  lines.  To 
this  end  he  defines  two  equidistant  straight  lines  as  parallels, 
and  attempts  to  prove  that  the  locus  of  the  points  equidistant 
from  one  straight  line  is  another  straight  line.^ 

His  demonstration  practically  depends  upon  the  follow- 
ing lemma: 

1/  two  points,  A,  C  itpo7i  a  curve,  wJwse  concavity  is  to- 
7vards  X,  are  joined  by  the  straight  line  AC,  and  perpendiculars 
are  drawn  from  the  infinite  number  of  points  of  the  arc  AC 
upon  atiy  straight  line,  theft  these  perpendiculars  cannot  be  equal 
to  each  other. 

The  words  'any  straight  line',  in  this  enunciation,  do  not 

refer  to  a  straight  line  taken  at  random  in  the  plane,  but  to 

Q  P  a  straight  line  constructed  in 

the  following  way  (Fig.  6). 
From  the  point  B  of  the  arc 
AC  draw  BD  perpendicular  to 


^^A  D  ~C^   the  chordae.   Then  at  ^  draw 

^'s-  ^-  AG  also  perpendicular  to  AC. 

Finally,  having  cut  off  equal  segments  AG  and  DF  upon 
these  two  perpendiculars,  join  the  ends  G  and  F.  GF  is  the 
straight  line  which  Giordano  considers  in  his  demonstration, 
a  straight  line  with  respect  to  which  the  arc  AB  is  certainly 
not  an  equidistant  line. 

But  when  the  author  wishes  to  prove  that  the  locus  of 
points  equidistant  from  a  straight  line  is  also  a  straight  line^ 
he  applies  the  preceding  lemma  to  a  figure  in  which  the  re- 
lations existing  between  the  arc  ABC  and  the  straight  line 


I  Giordano  Vitale:  Euclide  restiluio  overo  gli  antichi  elementi 
geometrici  ristaurati.  e  facilitati.     Libri  XV.     (Rome,   1680). 


Giordano  Vitale's  Proof. 


15 


GF  do  not  hold.  Thus  the  consequences  which  he  deduces 
from  the  existence  of  equidistant  straight  lines  are  not  really 
legitimate. 

From  this  point  of  view  Giordano's  proof  makes  no  ad- 
vance upon  those  which  preceded  it.  However  it  includes  a 
most  remarkable  theorem,  containing  an  idea  which  will  be 
further  developed  in  the  articles  which  follow. 

Let  ABCD  be  a  quadrilateral  of  which  the  angles  A,  B 
are  right  angles  and  the  sides  AD,  BC 
equal  (Fig.  7).  Further,  XqIHK  be  the  per- 
pendicular drawn  from  a  point  H,  upon  the 
side  DC,  to  the  base  AB  of  the  quadri- 
lateral. Giordano  proves:  (ij  that  the  ang- 
les D,  C  are  equal;  (ii)  that,  when  the  seg- 
ment HK  is  equal  to  the  segment  AD,  the 
two  angles  D,  C  are  right  angles,  and  CD  is  equidistant 
from  AB. 

By  means  of  this  theorem  Giordano  reduces  the  question 
of  equidistant  straight  lines  to  the  proof  of  the  existence  of 
one  point  H  vc^QXi  DC,  whose  distance  from  AB  is  equal  to 
the  segments  AD  and  BC.  We  regard  this  as  one  of  the 
most  noteworthy  results  in  the  theory  of  parallels  obtained 
up  to  that  date.^ 

§  g.  J.  Wallis  [1616 — 1703]  abandoned  the  idea  of 
equidistance,  employed  without  success  by  the  preceding 
mathematicians,  and  gave  a  new  demonstration  of  the  Fifth 
Postulate.  He  based  his  proof  on  the  Axiom:  To  every  figure 
there  exists  a  similar  figure  of  arbitrary  viagnitude.  We  now 
describe  shortly  how  Wallis  proceeds:^ 

1  Cf.  :  BoNOLA:  Uti  teo?-e?na  di  Giordano  Vitale  da  Bitonto  sidle 
rette  equidistanti.  Bollettino  di  Bibliografia  e  Storia  delle  Scienze 
Mat.  (1905). 

2  Cf.  :  Wallis  :  De  Postulato  Quinto;  et  Definizione  Quinta;  Lib.  6. 


1 6        I.     The  Attempts  to  prove  Euclid's  Parallel  Postulate. 


Let  a,  b  be  two  straightlines  intersected  at  A^  B  by  the 
transversal  c  (Fig.  8).    Let  a,  p  be  the  interior  angles  on  the 

same  side  of  c,  such  that  a  +  p  is 
less  than  two  right  angles.  Through 
A  draw  the  straight  line  b'  so  that 
b  and  b'  form  with  c  equal  corre- 
sponding angles.  It  is  clear  that 
b'  will  he  in  the  angle  adjacent  to 
a.  Let  the  line  b  be  now  moved 
continuously  along  the  segment 
AB^  so  that  the  angle  which  it 
makes  with  c  remains  always  equal  to  p.  Before  it  reaches 
its  final  position  b'  it  must  necessarily  intersect  a.  In  this  way 
a  triangle  AB^C^  is  determined,  with  the  angles  at  A  and  B^ 
respectively  equal  to  a  and  p. 

But,  by  Wallis's  hypothesis  of  the  existence  of  similar 
figures,  upon  AB,  the  side  homologous  to  AB^,  we  must  be 
able  to  construct  a  triangle  ABC^wcAzx  to  the  triangle  AB^  Ci. 
This  is  equivalent  to  saying  that  the  straight  lines  a,  b  must 
meet  in  a  point,  namely,  the  third  angular  point  of  the  triangle 
ABC.   Therefore,  etc. 

Wallis  then  seeks  to  justify  the  new  position  he  has  taken 
up.  He  points  out  that  Euclid,  in  postulating  the  existence 
of  a  circle  of  given  centre  and  given  radius,  [Post.  III.],  practi- 
cally admits  the  principle  of  similarity  for  circles.  But  even 
although  intuition  would  support  this  view,  the  idea  of  form, 
independent  of  the  dimensions  of  the  figure,  constitutes  a 

Eiididis;  disceptatio  geometrica.  Opera  Math.  t.  II;  p.  669 — 78  (Oxford, 
1693).  This  work  by  Wallis  contains  two  lectures  given  by  him  in 
the  University  of  Oxford;  the  first  in  1651,  the  second  in  1663.  It 
also  contains  the  demonstration  of  Nasìr-Eddìn.  The  part  containing 
Wallis's  proof  was  translated  into  German  by  Engel  and  StAckel  in 
their  Theorie  der  ParaUellmien  von  Euclid  bis  auf  Gauss,  p.  21 — 36, 
(Leipzig,  Teubner,  1895).  We  shall  quote  this  work  in  future  as 
Th.  der  P. 


Wallis's  Proof. 


17 


hypothesis,  which  is  certainly  not  more  evident  than  the  Postu- 
late of  Euclid. 

We  remark,  further,  that  Wallis  could  more  simply  have 
assumed  the  existence  of  triangles  with  equal  angles,  or,  as 
we  shall  see  below,  of  only  two  unequal  triangles  whose 
angles  are  correspondingly  equal. 

[cf.  p.  29  Note  I.] 

§  10  .  The  critical  work  of  the  preceding  geometers  is 
sufficient  to  show  the  historical  development  of  our  subject  in 
the  i6tb  and  17th  Centuries,  so  that  it  would  be  superfluous 
to  speak  of  other  able  writers,  such  as,  e.  g.,  Oliver  of 
Bury  [1604],  Luca  Valerio  [1613],  H.  Savile  [1621], 
A.  Tacquet  [1654],  A.  Arnauld  [1667].^  However,  it  seems 
necessary  to  say  a  few  words  on  the  question  of  the  position 
which  the  different  commentators  on  the  '' Ele7nents'  allot  to 
the  Euclidean  hypothesis  in  the  system  of  geometry. 

In  the  Latin  edition  of  the  "" Elements'  [1482],  based  upon 
the  Arabian  texts,  by  Campanus  [13th  Century],  this  hypothesis 
finds  a  place  among  the  postulates.  The  same  may  be  said 
of  the  Latin  translation  of  the  Greek  version  by  B.  Zamberti 
[1505],  of  the  editions  of  Luca  Paciuolo  [1509],  of  N.  Tar- 
taglia [1543],  of  F.  Commanding  [1572],  and  of  G.  A.Bor- 
ELLi  [1658]. 

On  the  other  hand  the  first  printed  copy  of  the  'Ele- 
ments' in  Greek,  [Basle,  1533],  contains  the  hypothesis  among 
the  axioms  [Axiom  XI].  In  succession  it  is  placed  among  the 
Axioms  by  F.  Candalla  [1556],  C.  S.  Clavio  [1574],  Gior- 
dano Vitale  [1680],  and  also  by  Gregory  [1703],  in  his 
well-known  Latin  version  of  Euclid's  works. 

To  attempt  to  form  a  correct  judgment  upon  these  dis- 


I  For  fuller  information  on  this  subject  cf.  Riccardi:  Saggio 
di  una  bibliografia  euclidea.  Mem.  di  Bologna,  (5)  T.  I.  p.  27 — 34, 
(1890). 


1 8        I-     The  Attempts  to  prove  Euclid's  Parallel  Postulate. 

crepancies^  due  more  to  the  manuscripts  handed  down  from 
the  Greeks  than  to  the  aforesaid  authors,  it  will  be  an  advan- 
tage to  know  what  meaning  the  former  gave  to  the  words 
'postulates'  [aÌTniaara]  and  'axioms'  [dHid))uara].'  First  of  all 
we  note  that  the  word  ^axioms'  is  used  here  to  denote  what 
Euclid  in  his  text  calls  '"commcni  notions'  [KOivai  evvoiai]. 

Proclus  gives  three  different  ways  of  explaining  the  differ- 
ence between  the  axioms  and  postulates. 

The  first  method  takes  us  back  to  the  difference  between 
a  problem  and  a  theorem.  A  postulate  differs  from  an  axiom, 
as  a  problem  differs  from  a  theorem,  says  Proclus.  By  this  we 
must  understand  that  a  postulate  affirms  the  possibility  of  a 
construction. 

The  second  method  consists  in  saying  \\\2X  a  postulate  is 
a  proposition  with  a  geometrical  meaning,  while  an  axiom  is  a 
propontio7i  common  both  to  geometry  and  to  arithmetic. 

Finally  the  third  method  of  explaining  the  difference 
between  the  two  words,  given  by  Proclus,  is  supported  by  the 
authority  of  Aristotle  [384 — 322  B.  C.].  The  words  axiom 
2.\\^ postulate  à.0  not  appear  to  be  used  by  Aristotle  exclusive- 
1\-  in  the  mathematical  sense.  An  axiom  is  that  which  is  true 
in  itself,  that  is,  owing  to  the  meaning  of  the  words  which  it 
contains;  a  postulate  is  that  7vhich,  although  it  is  not  an  axiom, 
in  the  aforesaid  sense,  is  admitted  without  demonstration. 

Thus  the  word  axiom,  as  is  more  evident  from  an  ex- 
ample due  to  Aristotle,  \7i)hen  equal  things  are  subtracted  from 
equal  things  the  remainders  are  equal\  is  used  in  a  sense  which 

I  For  the  following,  cf.  Proclus,  in  the  chapter  entitled  Pe- 
ata et  axiomata.  In  a  Paper  read  at  the  Third  Mathematical  Congress 
(Heidelberg,  1904)  G.  Vailati  has  called  the  attention  of  students 
anew  to  the  meaning  of  these  words  among  the  Greeks.  Cf.  :  In- 
torno al  significato  della  distinzione  tra  gli  assiotiii  ed  i  postulati  nella 
geometria  greca.  Verh.  des  dritten  Math.  Kongresses,  p.  575 — 5^'» 
(Leipzig,  Teubner,  1005). 


Position  of  the  Parallel  Postulate.  jg 

corresponds,  at  any  rate  very  closely,  to  that  of  the  common 
notions  of  Euclid,  whilst  the  word  postulate  in  Aristotle  has 
a  different  meaning  from  each  of  the  two  to  which  reference 
has  just  been  made.' 

Hence  according  as  one  or  other  of  these  distinctions  be- 
tween the  words  is  adopted,  a  particular  proposition  would  be 
placed  among  the  postulates  or  among  the  axioms.  If  we 
adopt  the  first,  only  the  first  three  of  the  five  postulates  of 
Euclid,  according  to  Proclus,  have  a  right  to  this  name,  since 
only  in  these  are  we  asked  to  carry  out  a  construction  [to 
join  two  points,  to  produce  a  straight  line,  to  describe  a  circle 
v.'hose  centre  and  radius  are  arbitrary].  On  the  other  hand, 
Postulate  IV.  [all  right  angles  are  equal],  and  Postulate  V.  ought 
to  be  placed  among  the  axioms.* 

1  Cf.  Aristotle:  Analytica  Posteriora.  I,  lo.  §  8.  We  quote  in 
full  this  slightly  obscure  passage,  where  the  philosopher  speaks  of 
the  postulate:  6aa  fièv  ouv  beiKTÙ  òvxa  \a|updvei  aÙTÒq  \ì.t\  òeiEa^, 
TaÙTO  éàv  nèv  òokoOvto  \a|Lipdvr]  tuj  |aav6dvovTi  ÙTT0TÌ9eTai.  Kaì 
éariv  oùx  à-rrXuJq  ÙTTÓGeoK;  àWà  irpò?  éKeivov  fióvov.  'Eàv  bè  f) 
firibeiuià?  évoùjriq  òò^r\ii  f)  koì  évavTia(;  évouariq  \a.\\.^6.yix\,  tò  auro 
aÌTeìrm.  Kaì  toùtu;  òiaq)épei  OiróBean;  koì  airrìiuo,  ?(Jti  yàp 
aitrina  tò  ÙTrevavTi'ov  toO  juavGdvovxoq  Tf)  bóEr). 

2  It  is  right  to  remark  that  the  Fifth  Postulate  can  be  enun- 
ciated thus  :  The  common  point  of  two  straight  lines  can  be  found,  when 
these  two  lines,  cut  by  a  transversal,  form  two  interior  angles  on  the 
same  side  whose  sum  is  less  than  two  right  angles.  Thus  it  follows 
that  this  postulate  affirms,  like  the  first  three,  the  possibility  of  a 
construction.  However  this  character  disappears  altogether,  if  it 
is  enunciated,  for  example,  thus  :  Through  a  point  there  passes  only 
one  parallel  to  a  straight  line;  or,  thus  :  Two  straight  lines  which  are 
parallel  to  a  third  km  are  parallel  to  each  other.     It  would  therefore 

appear  that  the  distinction  noted  above  is  purely  formal.  However 
we  must  not  let  ourselves  be  deceived  by  appearances.  The  Fifth 
Postulate,  in  whatever  way  it  is  enunciated,  practically  allows  the 
construction  of  the  point  of  intersection  of  all  the  straight  lines  of 
a  pencil  with  a  given  straight  line  in  the  plane  of  the  pencil,  one 
of  these  lines  alone  being  excepted.     It  is  true  that  there  is  a  certain 


20        I-     The  Attempts  to  prove  Euclid's  Parallel  Postulate. 

Again,  if  we  accept  the  second  or  the  third  distinction, 
the  five  Euclidean  postulates  should  all  be  included  among 
the  postulates. 

In  this  way  the  origin  of  the  divergence  between  the  var- 
ious manuscripts  is  easily  explained.  To  give  greater  weight 
to  this  explanation  we  might  add  the  uncertainty  which  histor- 
ians feel  in  attributing  to  Euclid  the  postulates,  common  no- 
tions and  definitions  of  the  First  Book.  So  tar  as  regards  the 
postulates,  the  gravest  doubts  are  directed  against  the  last 
two.  The  presence  of  the  first  three  is  sufficiently  in  accord 
with  the  whole  plan  of  the  work.'  Admitting  the  hypothesis 
that  the  Fourth  and  Fifth  Postulates  are  not  Euclid's,  even  if 
it  is  against  the  authority  of  Geminus  and  Proclus,  the  ex- 
treme rigour  of  the  ''Elements''  would  naturally  lead  the  later 
geometers  to  seek  in  the  body  of  the  work  all  those  pro- 
positions which  are  admitted  without  demonstration.  Now 
the  one  which  concerns  us  is  found  stated  very  concisely  in 
the  demonstration  of  Bk.  I.  Prop.  29.  From  this,  the  sub- 
stance of  the  Fifth  Postulate  could  then  be  taken,  and  added 
to  the  postulates  of  construction,  or  to  the  axioms,  according 
to  the  views  held  by  the  transcriber  of  Euclid's  work. 

Further,  its  natural  place  would  be,  and  this  is  Gregory's 
view,  after  Prop.  27,  of  which  it  enunciates  the  converse. 

Finally,  we  remark  that,  whatever  be  the  manner  of  de- 
ciding the  verbal  question  here  raised,  the  modern  philo- 
sophy of  mathematics  is  inclined  generally  to  suppress  the 


difference  between  this  postulate  and  the  three  postulates  of  con- 
struction. In  the  latter  the  data  are  completely  independent.  In 
the  former  the  data  (the  two  straight  lines  cut  by  a  transversal)  are 
subject  to  a  condition.  So  that  the  Euclidean  Hypothesis  belongs 
to  a  class  intermediate  between  the  postulates  and  axiom,  rather 
than  to  the  one  or  the  other. 

I   Cf.  P.  Tannery:  Sur  Pauthentuité  des  axiomes  d'Euclide.    Bull, 
d.  Sc.  Math.  (2),  T.  VIII.  p.    162—175,  (1884). 


Postulates  and  Axioms.  21 

distinction  between  postulate  and  axiom,  which  is  adopted  in 
the  second  and  third  of  the  above  methods.  The  generally 
accepted  view  is  to  regard  the  fundamental  propositions  of 
geometry  as  hypotheses  resting  upon  an  empirical  basis, 
while  it  is  considered  superfluous  to  place  statements,  which 
are  simple  consequences  of  the  given  definitions,  among  the 
propositions. 


Chapter  IL 

The  Forerunners  of  Non-Euclidean 
Geometry. 

Gerolamo  Saccheri  [1667 — 1733]. 

§  II.  The  greater  part  of  the  work  of  Gerolamo  Sac- 
cheri: EucHdes  ab  o/nni  Jiaevo  vindicatus  :  sive  conatus  gco- 
meiricus  quo  stabiliuntur  prima  ipsa  universae  Geoinetriae 
Principia,  [Milan,  1733],  is  devoted  to  the  proof  of  the  Fifth 
Postulate.  The  distinctive  feature  of  Saccheri's  geometrical 
writings  is  to  be  found  in  his  ^Logica  de/i:o?!strativa' ,  [Turin, 
1697J.  It  is  simply  a  particular  method  of  reasoning,  already 
used  by  Euclid  [Bk.  IX.  Prop.  1 2J,  according  to  which  by 
assuming  as  hypothesis  that  the  proposition  7vhlch  is  to  beproi  ed 
is  false,  one  ts  brought  to  the  conclusion  that  it  is  true} 

Adopting  this  idea,  the  author  takes  as  data  the  first 
twenty-six  propositions  of  Euclid,  and  he  assumes  as  a  hypo- 
thesis that  the  Fifth  Postulate  is  false.  Among  the  consequences 
of  this  hypothesis  he  seeks  for  some  proposition,  which  would 
entitle  him  to  affirm  the  truth  of  the  postulate  itself. 

Before  entering  upon  an  exposition  of  Saccheri's  work, 
we  note  that  Euclid  assumes  implicitly  that  the  straight  line 
is  infinite  in  the  demonstration  of  Bk,  I.  16  [the  exterior  angle 
of  a  triangle  is  greater  than  either  of  the  interior  and  opposite 

'  Cf.  G.  Vailati:  Di  iin^  o/era  dimenlicata  del  P.  Gerolamo  Sac- 
chtri.  Rivista  Filosofica  (1903). 


Saccheri's  Quadrilateral.  23 

angles],  since  his  argument  is  practically  based  upon  the 
existence  of  a  segment  which  is  double  a  given  segment. 

We  shall  deal  later  with  the  possibihty  of  abandoning 
this  hypothesis.  At  present  we  note  that  Saccheri  tacitly  as- 
sumes it,  since  in  the  course  of  his  work  he  uses  Xh^ proposition 
of  the  exterior  angle. 

Finally,  we  note  that  he  also  employs  the  Postulate  of 
Archimedes^  and  the  hypothesis  of  the  continuity  of  the  straight 
liae,^  to  extend,  to  all  the  figures  of  a  given  type,  certain  pro- 
positions admitted  to  be  true  only  for  a  single  figure  of  that 
type. 

§  12.  The  fundamental  figure  of  Saccheri  is  the  two 
right-ar.gled  isosceles  quadrilateral;  that  is,  the  quadrilateral  of 
which  two  opposite  sides  are  equal  to  each  other  and  perpen- 
dicular to  the  base.  The  properties  of  such  a  figure  are  de- 
duced from  the  following  Lemma  I. ,  which  can  easily  be 
proved  : 

If  a  quadrilateral  ABCD  has  the  consecutive  angles  A 
a7id  B  right  angles,  and  the  sides  AD  and  BC  equal,  then  the 
angle  C  is  equal  to  the  angle  D  [This  is  a  special  case  of  Sac- 
cheri's Prop.  I.];  but  if  the  sides  AD  and  BC  are  unequal,  of 
the  two  angles  C,  Z>,  that  one  is  greater  which  is  adjacent  to 
the  shorter  side,  and  vice  versa. 


1  [The  Postulate  of  Archimedes  is  stated  by  Hilbert  thus:  Let 
Al  be  any  point  upon  a  straight  line  between  the  arbitrarily  chosen 
points  A  and  B.  Take  the  points  A2,  A^,  .  .  .  so  that  Ai  lies 
between  A  and  A2,  A2  between  Ai  and  /i;„  etc.;  moreover  let  the 
segments  AAi,  A1A2,  ^2^3,  ...  be  all  equal.  Then  among  this 
series  of  points,  there  always  exists  a  ceitain  point  Ad,  such  that 
B  lies  between  A  and  Aa-\ 

2  This  hypothesis  is  used  by  Saccheri  in  its  intuitive  form, 
viz.  :  a  segment,  which  passes  continuously  from  the  length  a  to 
the  length  b,  different  from  a,  takes,  during  its  variation,  every 
length  intermediate  between  a  and  b. 


24 


II.     The  Forerunners  of  Non-Euclidean  Geometry. 


Let  ABCD  be  a  quadrilateral  with  two  right  angles  A 
and  B,  and  two  equal  sides  AD  and  BC  (Fig.  9).  On  the 
Euclidean  hypothesis  the  angles  Cand  D  are  also  right  angles. 
Thus,  if  we  assume  that  they  are  able  to  be  both  obtuse,  or 
both  acute,  we  implicitly  deny  the  Fifth  Postulate.  Saccheri 
discusses  these  three  hypotheses  regarding  the  angles  C,  D. 
He  named  them: 

The  Hypothesis  of  the  Right  Angle 

[<^  6"=  <^  Z>  =  I  right  angle]  : 
The  Hypothesis  of  the  Obtuse  A/igle 

[-^  C  =  <^  Z>  >  I  right  angle]  : 
The  Hypothesis  of  the  Acute  Angle 

[^  C=  <^Z>  <  I  right  angle]. 
One  of  his  first  important  results  is  the  following: 
Accordifig  as  the  Hypothesis  of  the  Eight  Angle,  of  the 
Obtuse  Angle,  or  of  the  Acute  Attgle  is  true  i?i  the  two  right- 
angled   isosceles    quadrilateral,    we    must    have  AB  =  CD, 
ABy-  CD,  or  AB  <  CD,  respectively.    [Prop.  111.] 

In  fact,  on  the  Hypothesis  of  the  Eight  Angle,  by  the 
preceding  Lemma,  we  have  immediately 
AB  =  CD. 
On  the  Hypothesis  of  the  Obtuse  Angle,  the  perpendicular 
00'  at  the  middle  point  of  the  segment  y^/>' 
divides  the   fundamental   quadrilateral  into 
two  equal  quadrilaterals,  with  right  angles  at 
O  and   O'.     Since  the  angle  D  ^  angle  A, 
then  we   must   have   AO  ^  DO ,   by  this 
Lemma.    Thus  AB  >  CD. 

On  the  Hypothesis  of  the  Acute  Angle  these 
^'^  9-  inequalities  have  their  sense  changed  and 

we  have 

AB  <  CD. 
Using  the  reductio  ad  absurdum  argument,  we  obtain 
the  converse  of  this  theorem.   [Prop.  IV.] 


O 


•The  Three  Hypotheses.  25 

If  the  Hypothesis  of  the  Right  Angle  is  true  in  only  one 
4ase,  then  it  is  true  in  every  other  case.    [Prop.  V.] 

Suppose  that  in  the  two  right-angled  isosceles  quadrilat- 
eral AB  CD  the  Hypothesis  of  the  Eight  Angle  is  verified. 

In  AD  and  BC  (Fig.  lo)  take  the  points  ZTand  K  equi- 
distant from  AB;  join  HK  a.nd  form  the 


P 


quadrilateral  ABKH.  m 

If  HK  is  perpendicular  to  AH  and 
BK,  the  Hypothesis  of  the  Right  Angle  is    ^^j- 
also  verified  in  the  new  quadrilateral.  H 

If  it  is  not,  suppose  that  the  angle 
AHK  is  acute.   Then  the  adjacent  angle       ^  ^ 

DHK  is  obtuse.  Thus  in  the  quadrilateral  '^'  ^  ' 

ABKH,  from  the  Hypothesis  of  the  Acute  Angle,  it  follows 
that  AB  <C  HK:  while  in  the  quadrilateral  HKCD,  from  the 
Hypothesis  of  the  Obtuse  Angle,  it  follows  that  HK<^  CD. 

But  these  two  inequalities  are  contradictory,  since  by 
4he  Hypothesis  of  the  Right  Angle  in  the  quadrilateral  ABCD, 
AB  =  CD. 

Thus  the  angle  AHK  cannot  be  acute  :  and  since  by  the 
same  reasoning  we  could  prove  that  the  angle  AHK  cannot 
be  obtuse,  it  follows  that  the  Hypothesis  of  the  Right  Angle  is 
also  true  in  the  quadrilateral  ABKH. 

On  AD  and  i)C  produced,  take  the  points  M,  iV  equi- 
distant from  the  base  AB.  Then  the  Hypothesis  of  the  Right 
Angle  is  also  true  for  the  quadrilateral  AB  JVM.  In  fact  if 
AM  is  a  multiple  of  AD,  the  proposition  is  obvious,  li  AM 
is  not  a  multiple  of  AD,  we  take  a  multiple  of  AD  greater 
than  AM  \the  Postulate  of  Archimedes^  and  from  AD  and 
BC  produced  cut  off  AF  and  BQ  equal  to  this  multiple. 
Since,  as  we  have  just  seen,  the  Hypothesis  of  the  Right  Angle 
is  true  in  the  quadrilateral  ABQF,  the  same  hypothesis  must 
also  hold  in  the  quadrilateral  ABNM. 

Finally  the  said  hypothesis  must  hold  for  a  quadrilateral 


26  II-     The  Forerunners   of  Non-Euclidean  Geometry. 

on  any  base,  since,  in  Fig.  lo,  we  can  take  as  the  base  one 
of  the  sides  perpendicular  to  AB. 

Note.  This  theorem  of  Saccheri  is  practically  contained 
in  that  of  Giordano  Vitale,  stated  on  p.  15.  In  fact,  refer- 
ring to  Fig.  7,  the  hypothesis 

DA==  HK^  CR 
is  equivalent  to  the  other 

<5C  Z>  =  -^  H=-  <  C  =  I  right  angle. 
Ikit  from  the  former,  there  follows  the  equidistance  of  the 
two  straight  lines  DC,  AB^;  and  thus  the  validity  of  the  Hypo- 
thesis of  the  Right  Angle  in  all  the  two  right-angled  isosceles 
quadrilaterals,  whose  altitude  is  equal  to  the  line  DA,  is 
established.  The  same  hypothesis  is  also  true  in  a  quadri- 
lateral of  any  height,  since  the  line  called  at  one  time  the 
base  may  later  be  regarded  as  the  height. 

If  the  Hypothesis  of  the  Obtuse  Angle  is  true  in  only  one 
case,  then  it  is  true  in  every  other  case.    [Prop.  VI.] 

Referring  to  the  standard  quadrilateral  ^j9CZ>  (Fig.  11), 
n      K 1         C        suppose  that  the  angles  C  and  D  are  ob- 
tuse.   Upon  AD  and  BC  take  the  points 
H  and  K  equidistant  from  AB. 

In  the  first  place  we  note  that  the 

segment  HK  cannot  be  perpendicular  to 

the  two  sides  AD  and  BC,  since  in  that 

A       Oj  B      case  the  Hypothesis  of  the  Right  Angle 

Fig.  II.  would    be    verified    in   the   quadrilateral 

ABKH,  and  consequently  in  the  fundamental  quadrilateral. 

Let  us  suppose  that  the  angle  AHK  is   acute.    Then 


I  It  is  true  that  Giordano  in  his  argument  refers  to  the  points 
of  the  segment  DC,  which  he  shows  are  equidistant  from  the  base 
AB  of  the  quadrilateral.  However  the  same  argument  is  applicable 
to  all  the  points  which  lie  upon  DC,  or  upon  DC  produced.  Cf. 
Bonola's  Note  referred  to  on  p.   15. 


Proof  for  one  Quadrilateral  Sufficient.  27 

by  the  Hypothesis  of  the  Acute  Angle,  HK^  AB.  But  as  the 
Hypothesis  of  the  Obtuse  Angle  holds  in  AB  DC,  we  have 

AB^  CD. 
Therefore  HK^  AB  >  CD. 

If  we  now  move  the  straight  line  HK  continuously,  so  that  it 
remains  perpendicular  to  the  median  00'  of  the  fundamental 
quadrilateral,  the  segment  HK,  contained  between  the  oppo- 
site sides  AD,  BC,  which  in  its  initial  position  is  greater  than 
AB,  will  become  less  than  AB  in  its  final  position  DC.  From 
the  postulate  of  continuity  we  may  then  conclude  that, 
between  the  initial  position  HK  and  the  final  position  DC, 
there  must  exist  an  intermediate  position  H' K' ,  for  which 
H'K'  =  AB. 

Consequently  in  the  quadrilateral  ABK'H'  the  Hypo- 
thesis of  the  Right  Angle  would  hold  [Prop.  III.J,  and  therefore, 
by  the  preceding  theorem,  the  Hypothesis  of  the  Obtuse  Angle 
could  not  be  true  in  ABCD. 

The  argument  is  also  valid  if  the  segments  ^j^,  BK  arc 
greater  than  AD,  since  it  is  impossible  that  the  angle  AHK 
could  be  acute.  Thus  the  Hypothesis  of  the  Obtuse  Angle  holds 
in  ABKH  as  well  as  in  ABCD. 

Having  proved  the  theorem  for  a  quadrilateral  whose 
sides  are  of  any  size,  we  proceed  to  prove  it  for  one  whose 
base  is  of  any  size:  for  example  the  base  BK  [cf  Fig.  12]. 

Since  the  angles  K,  H,  are  obtuse,  the 
perpendicular  at  K  to  KB  will  meet  the 
segment  AH  in  the  point  M,  making  the 
angle  AMK  obtuse  [theorem  of  the  ex- 
terior angle]. 

Then  in  ABKM  we  have  AB  >  KM, 
by  Lemma  I.   Cut  off  from  AB  the  segment 
-5iV equal  to  MK.  Then  we  can  construct 
the  two  right  angled  isosceles  quadrilateral  BKMN,  with  the 
angle  MNB  obtuse,  since  it  is  an  exterior  angle  of  the  triangle 


28  II-     The  Forerunners  of  Non-Euclidean  Geometry. 

ANAL     It  follows  that  the  Hypothesis  of  the  Obtuse  A?igle 
holds  in  the  new  quadrilateral. 

Thus  the  theorem  is  completely  demonstrated. 

1/  the  Hypothesis  of  the  Acute  Angle  is  true  in  only  one 
case,  then  it  is  true  in  every  other  case.    [Prop.  VII.] 

This  theorem  can  be  easily  proved  by  using  the  method 
of  reductio  ad  absurdum. 

§  13.  From  the  theorems  of  the  last  article  Saccher: 
easily  obtains  the  following  important  result  with  regard  to 
triangles  : 

According  as  the  Hypothesis  of  the  Right  Angle,  the  Hy- 
pothesis of  the  Obtuse  Angle,  or  the  Hypothesis  of  the  Acute 
Angle,  is  found  to  be  true,  the  sum  of  the  angles  of  a  triangle 
will  be  respectively  equal  to,  greater  than^  or  less  thafi  two  right 
angles.    [Prop.  IX.] 

Let  ABC  [Fig.  1 3]  be  a  triangle  of  which  ^  is  a  right 
P    angle.    Complete  the  quadrilateral  by  draw- 
ing AD  perpendicular  to  AB  and  equal  to 
BC;  and  jon  CD. 

On  the  Hypothesis  of  the  Bight  Afigle, 
the  two  triangles  ABC  and  ADC  are  equal. 
Therefore  -^BAC^^^DCA. 
It  follows  immediately  that  in  the  tri- 
angle ABC, 
^A-\-  ^B  +  <^  C=  2  right  angles. 
On  the  Hypothesis  of  the  Obtuse  Angle, 
sinc^AB^DC, 
we  have  ^ACB^  <C  DAC.  ' 


I  This  inequality  is  proved  by  Saccheri  in  his  Prop.  VIII., 
and  serves  as  Lemma  to  Prop.  IX.  It  is,  of  course.  Prop.  25  of 
Euclid's  First  Book. 


The  Sum  of  the  Angles  of  a  Triangle.  2Q 

Therefore,  in  this  triangle  we  shall  have 

■^  A  +  ^J5  +  -^  C^  2  right  angles. 

On  the  Hypothesis  of  the  Acute  Angle, 
since  AB<^DC, 
we  have  ^ACB<C  ^£>AC, 
and  therefore,  in  the  same  triangle, 

<CA-\-  <^+  <:C<2  right  angles. 
The   theorem  just  proved   can  be  easily  extended    to   the 
case  of  any  triangle,  by  breaking  the  figure  up  into  two  right 
angled  triangles.  In  Prop.  XV.  Saccheri  proves  the  converse, 
by  a  reductio  ad  absurdum. 

The  following  theorem  is  a  simple  deduction  from  these 
results  : 

If  the  sum  of  the  angles  of  a  triangle  is  equal  to,  greater 
than,  or  less  than  two  right  at/gles  iti  only  one  triangle,  this 
sum  will  be  respectively  equal  to,  greater  than,  or  less  than  t7vo 
right  angles  in  every  other  triangle.'^ 

This  theorem,  which  Saccheri  does  not  enunciate  ex- 
plicitly, Legendre  discovered  anew  and  published,  for  the 
first  and  third  hypotheses,  about  a  century  later. 

§  14.  The  preceding  theorems  on  the  two  right- 
angled  isosceles  quadrilaterals  were  proved  by  Saccheri,  and 


I  Another  of  Saccheri's  propositions,  which  does  not  concern 
us  directly,  states  that  if  the  sum  of  the  angles  of  only  one  quadri- 
lateral is  equal  to,  greater  than,  or  less  than  four  right  angles,  the 
Hypothesis  of  the  Right  Angle,  the  Hypothesis  of  the  Obtuse  Angle,  or 
the  Hypothesis  of  the  Acute  Angle  zvould  respectively  be  true.  A  note 
of  Saccheri's  on  the  Postulate  of  Wallis  (cf.  %  9)  makes  use  of 
this  proposition.  He  points  out  that  Wallis  needed  only  to  assume 
the  existence  of  two  triangles,  whose  angles  were  equal  each  to 
each  and  sides  unequal,  to  deduce  the  existence  of  a  quadrilateral 
in  which  the  sum  of  the  angles  is  equal  to  four  right  angles.  From 
this  the  validity  of  the  Hypothesis  of  the  Right  Angle  would  follow, 
and  in  its  turn  the  Fifth  Postulate. 


■JO  II-     The  Forerunners  of  Non-Euclidean  Geometry. 

later  by  other  geometers^  with  the  help  oi  ^t  Postulate  of 
Anhimedes  and  the  principle  of  contifiuity  [cf.  Prop.  V.,  VI]. 
However  Dehn^  has  shown  that  they  are  independent  of 
these  hypotheses.  This  can  also  be  proved  in  an  elementary 
way  as  follows.^ 

On  the  straight  line  r  (Fig.  14)  let  two  points  B  and  D 
be  chosen,  and  equal  perpendiculars  BA  and  DC  be  drawn 
to  these  lines.  Let  A  and  C  be  joined  by  the  straight  line  s. 
The  figure  so  obtained,  in  which  evidently  <fiBAC=  -^  Z>CA, 
is  fundamental  in  our  argument  and  we  shall  refer  to  it  con- 
stantly. 

Two  points  Jt,  E'  are  now  taken  on  j,  of  which  the 
first  is  situated  between  A  and  C,  and  the  second  not. 

Further  let  the  perpendiculars  from  E^  E'  to  the  line 
r  meet  it  at  E  and  E' . 

The  following  theorems  now  hold: 

\\{  EE^AB,\ 
I.    I  or  L  the  angles  BAC^  DCA  areright  angles. 

I     E'E'  =  AB  J 

\\i  EE>AB,\ 
II.    '  or  ^,  the  angles  ^^C,  Z>6>y  are  obtuse. 

I    E'E'<iAB\ 

niEE<iAB,\ 
III.  or  i ,  the  angles  BAC,  DCA  are  acute. 

[    E'E'^AB] 

We  now  prove  Theorem  I.   [cf.  Fig.  14.] 

From  the  hypothesis  EE  =  AB,  the  following  equalities 
are  deduced: 

1  Cf.  Die  Legendreschen  Satze  iiber  die  IVinkehiimme  im  Dreieck. 
Math.  Ann.  Bd.  53,  p.  405 — 439  (1900). 

2  Cf.  BoNOLA,  /  teoremi  del  Padre  Gerolamo  Sacrheri  sulla 
somma  degli  angoli  di  111/  triangolo  e  le  ricerche  di  M.  Dehn,  Rend. 
Istituto  Lombardo  (2);  Voi.  \XX.VIII.  (1905). 


Postulate  of  Archimedes  not  needed. 


31 


<^  BAE  =  ^  FEA,  and  <C  FEC  =  <r  DCE. 
These,  together  with  the  fundamental  equality 

^BAC=^DCA, 
are  sufficient  to  establish  the  equality  of  the  two  angles  FEA 
and  FEC. 

E      A         E        C 


—  s 


F'     B 


F        D 

Fig.  14. 


Since  these  are  adjacent  angles,  they  are  both  right 
angles,  and  consequently  the  angles  BAC  and  DCA  are 
right  angles. 

The  same  argument  is  applicable  in  the  hypothesis 
E'F'  =  AB. 

We  proceed  to  Theorem  11  [cf  Fig.  1 5]. 


Suppose,  in  the  first  place,  EF  >  AB.    From  FE  cut 
off  j^/=  AB,  and  join  /  to  A  and  C. 

Then  the  following  equalities  hold: 

^  BA/=  <è:  EIA  and  ^r  DCJ  ^  ^  FJC. 
Further,  by  the  theorem   of  the  exterior  angle  [Bk.  !.  16], 
we  have 


■22  n.     The  Forerunners  of  Non-Euclidean  Geometry. 

^  FIA  +  <^  FIC^  <^  FEA  +  <:  FEC  =  2  right  angles. 
But 

^BAC  ^  <^E>CA-><3Z£AI+  ^DCI. 
Therefore 

^BAC  ^  ^  DC  A  >  <:  FIA  +  <:  i^/C>  2  right  angles. 
But,  since  <  BAC^  <^  BCA, 
it  follows  that  -^BAC^  1  right  angle.  .  .  .  Q.  E.  D. 

In  the  second  place,  suppose  that  E'F'  <C  AB.    Then  from 

F'E'  produced  cut  off  F'F  =  BA,  and  join  /'  to  C  and  A. 

The  following  relations,  as  usual,  hold: 

^  i^'/'^  =  -^  BAT',  ^  FTC  =  <^  DCI'; 

^  /'^i5'  >  <^  rCE\  ^  F'rA<_  <ac  i^'/'C. 

Combining  these  results,  we  deduce,  first  of  all,  that 

^  BAI'<^  <C  ^C7'. 
From  this,  if  we  subtract  the  terms  of  the  inequality 

■^i'ae':><:J'ce', 

we  obtain 

<  BAE'<i  <:nCE'  =  ^  BAC. 
But  the  two  angles  BAE'  and  BAC  are  adjacent.     Thus  we 
have  proved  that  <C  BAC  is  obtuse. — Q.  E.  D. 

Theorem  III.  can  be  proved  in  exactly  the  same  way. 
The  converses  of  these  theorems  can  now  be  easily 
shown  to  be  true  by  the  reductio  ad  absurdum  method.  In 
particular,  if  M  and  N  are  the  middle  points  ot  the  two  seg- 
ments AC  and  BE>,  we  have  the  following  results  for  the 
segment  MN  which  is  perpendicular  to  both  the  hnes  AC 
and  BD  (Fig.  16). 

If  <r.  BAC=  r  nCA  =  /  right  angle,  then  MN=  AB. 
If  ^  BAC^-  ^DCA  >  /  right  angle,  then  AIN^AB. 
If  ^  BAC==  <^  nCA  <  /  right  angle,  then  MN<  AB. 
Further  it  is  easy  to  see  that 

(i)     If  <f^  BAC  =  <^  DCA  =  /  right  angle, 
then  <^  FEM  and  -^  F'E'M  are  each  i  right  angle. 


Bonola's  Proof. 


33 


(ii)  If  -^  BAC  ==  <:  nCA  '>  I  right  angle, 
then  <^  FEM  and  <f^  F' E' M  are  each  obtuse. 

(Hi)  If  <  BAC  =  <^  DC  A  <  I  right  angle, 
then  <^  FEM  and  -^  F'E'M  are  each  acute. 


A.     E 


^ 


F'    B     F 


In  fact,  in  Case  (i),  since  the  lines  r  and  s  are  equi- 
distant, the  following  equalities  hold: 
^NMA  =  ^FEM=^  <^  BAC=  ^F'E'M=i  right  angle. 

To  prove  Cases  (ii)  and  (iii),  it  is  sufficient  to  use  the 
reductio  ad  absurdum  method,  and  to  take  account  of  the 
results  obtained  above. 

Now  let  P  be  a  point  on  the  line  MN,  not  contained 
between  J/ and  iV(Fig.  1 7).  Let  RP  be  the  perpendicular  to 
MN  and  RK  the  perpendicular  to  BD.  This  last  perpend- 
icular will  meet  AC  in  a  point  H.  On  this  understanding 
the  preceding  theorems  immediately  establish  the  truth  of 
the  following  results: 

If-^BAM^i  right  angle,  then  <^  KHM and ^ KRF 
are  each  equal  to  i  right  angle. 

If  <^  BAAC>  r  right  angle,  then  <^  KHM  and  ^  KRF 
are  each  greater  than  i  right  angle. 

If  ^  BAM  <  /  right  angle,  then  <^  KHM  and  <^  KRF 
are  each  less  than  i  right  angle. 

These  results  are  also  true,  as  can  easily  be  seen,  if  the 
point  F  falls  between  M  and  N. 

In  conclusion,  the  last  three  theorems,  which  clearly 


34 


II.     The  Forerunners  of  Non-Euclidean  Geometry. 


coincide  with  Saccheri's  theorems  upon  the  two  right-angled 
isosceles  quadrilateral,  are  equivalent  to  the  following  result, 
proved  without  using  Archimedes'  Postulate:  — 


R 

P 

H 

A 

M         C 

K    B 

N         D 

If  the  truth  of  the  Hypothesis  of  the  Right  Angle,  of  the 
Obtuse  Angle,  or  of  the  Acute  Angle,  respectively,  is  known  in 
only  o?ie  case,  its  truth  is  also  kno2V7i  in  every  other  case. 

If  we  wish  now  to  pass  from  the  theorems  on  quad- 
rilaterals to  the  corresponding  theorems  on  triangles,  we  need 
only  refer  to  Saccheri's  demonstration  [cf.  p.  28],  since  this 
part  of  his  argument  does  not  in  any  way  depend  upon 
the  postulate  in  question. 

We  have  thus  obtained  the  result  which  was  to  be 
proved. 

§  15.  To  make  our  exposition  of  Saccheri's  work 
more  concise,  we  take  from  Prop.  XI.  and  XII.  the  contents 
of  the  following  Lemma  II: 

Let  ABC  be  a  tria?igle  of  ivhich  C  is  a  right  angle:  let 
H  be  the  tniddle  point  of  AB,  and  K  the  foot  of  the  perpen- 
dicular fro?n  H  upon  AC.    Then  we  shall  have 

AK  =  KC,  Oil  the  Hypothesis  of  the  Right  Angle; 

AK  <^  KC,  on  the  Hypothesis  of  the  Obtuse  Angle; 

AK  ^  KC,  on  the  Hypothesis  of  the  Acute  Angle. 

On  the  Hypothesis  of  the  Right  Angle  the  result  is 
obvious. 


The  Projection  of  a  Line. 


35 


On  the  Hypothesis  of  the  Obtuse  Angie,  since  the  sum  of 
the  angles  of  a  quadrilateral  is  greater  than  four  right  angles, 
it  follows  that  ^AHK <,^HBC.  Let  HL be  the  perpendi- 
cular from  H  to  BC  (Fig.  i8).  Then  the  result  just  obtained, 
and  the  fact  that  the  two  triangles  AHK,  HBL  have  equal 
hypotenuses,  give  rise  to  the  following  inequality  :  AK<^HL. 
But  the  quadrilateral  HKCL  has  three  right  angles  and  there- 
fore the  angle  H  is  obtuse  {Hypothesis  of  the  Obtuse  Angle], 
It  follows  that 

HL  <  KC, 
and  thus 

AK<^KC. 

The  third  part  of  this  Lemma  can  be  proved  in  the 
same  way. 

It  is  easy  to  extend  this  Lemma  as  follows  (Fig.  1 9)  : 


} 

Fig.  19. 

Lemma  LLI.  Lf  oti  the  one  arm  of  an  angle  A  equal  seg- 
ments AA^,  A-i_A^,  A^A.^y  .  .  .  are  taken,  and  AA^ ,  A^A^^ 
AjA^'. . .  are  their  projections  upon  the  other  arm  of  the  afigle, 
then  the  following  results  are  true: 

AAi  =  A-iA^  =  A^A^  =  .  .  . 
on  the  Hypothesis  of  the  Right  Angle; 

aa,'<:a,'a,'<a;a,  =  <. . . 

on  the  Hypothesis  of  the  Obtuse  Angle; 

aa;>a,'a/>a;a:>... 

on  the  Hypothesis  of  the  Acute  Angle. 

To  save  space  the  simple  demonstration  is  omitted. 

3* 


36 


II.     The  Forerunners  of  Non-Euclidean  Geometry. 


We  can  now  proceed  to  the  proof  of  Prop.  XI.  and  XII. 
of  Saccheri's  work,  combining  them  in  the  following  theorem: 

On  the  Hypothesis  of  the  Right  Angle  and  on  the  Hypo- 
thesis of  the  Obtuse  Angle,  a  line  perpendicular  to  a  given 
straight  line  and  a  lifie  cutting  it  at  an  acute  angle  intersect 
each  other. 


Fig.  ao. 


Let  (Fig,  2  o)  LP  and  AD  be  two  straight  lines  of  which 
the  one  is  perpendicular  to  AP,  and  the  other  is  inclined  to 
AP  at  an  acute  angle  DAP. 

After  cutting  off  in  succession  equal  segments  AD,  DF^, 
upon  AD,  draw  the  perpendiculars  DB  and  P^M^  upon  the 
line  AP. 

From  Lemma  III.  above,  we  have 
PM^  >  AB, 
or  AM^  ^  2  AB, 

on  the  two  hypotheses. 

Now  cut  off  Pip2  equal  to  AP^,  from  AP^  produced, 
and  let  M^  be  the  foot  of  the  perpendicular  from  P2  upon  AP. 
Then  we  have 

AM2  ^  2  AMi, 
and  thus 

AM2  >  2'  AB. 
This  process  can  be  repeated  as  often  as  we  please. 

In  this  way  we  would  obtain  a  point  Pu  upon  the  line 
AD  such  that  its  projection  upon  the  line  AP  would  deter- 
mine a  segment  A.if„  satisfying  the  relation 


Two  Hypotheses  give  Postulate  V. 


37 


AM"  >  2" AB. 

But  if  n  is  taken  sufficiently  great,  [by  the  Postulate  of 
Archimedes'^^  we  would  have 

2''  AB^AP, 
and  therefore 

AMn  >  AP. 
Therefore  the  point  P  lies  upon  the  side  AMn  of  the  right- 
angled  triangle  AM„  Fn-    The    perpendicular  PL  cannot 
intersect  the  other  side  of  this  triangle;  therefore  it  cuts  the 
hypotenuse.^     Q^.  E.  D. 

It  is  now  possible  to  prove  the  following  theorem  : 

T?ie  Fifth  Postulate  is  true  on  the  Hypothesis  of  the 
Right  Angle  at  id  on  the  Hypothesis  of  the  Obtuse  Angle  [Prop. 
XIII.]. 

Let  (Fig.  2i)  AB,  CD  be  two  straight  lines  cut  by  the 
line  AC. 

Let  us  suppose  that 

^  BAC  +  ^  ^CZ>  <  2  right  angles. 

Then  one  of  the  angles 
BAC,  ACD,  for  example  the 
first,  will  be  acute. 

From  C  draw  the  perpen- 
dicular CH  upon  AB.    In  the 
triangle  ACH,  from  the  hypo- 
theses which  have  been  made,    A 
we  shall  have 

<^A-\r  <^C  +  <C-^>2  right  angles. 


1  The  Postulate  of  Archimedes,  of  which  use  is  here  made, 
includes  implicitly  the  infinity  of  the  straight  line. 

2  The  method  followed  by  Saccheri  in  proving  this  theorem 
is  practically  the  same  as  that  of  Nasìr-Eddìn.  However  Nasir- 
Eddìn  only  deals  with  the  Hypothesis  of  the  Right  Angle,  as  he  had 
formerly  shown  that  the  sum  of  the  angles  of  a  triangle  is  equal 
to  two  right  angles.  It  is  right  to  remember  that  Saccheri  was 
familiar  with  and  had  criticised  the  work  of  the  Arabian  Geometer. 


28  II.     The  Forerunners  of  Non-Euclidean  Geometry. 

But  we  have  assumed  that 

<^  BAC  +  <^  ACD  <  2  right  angles. 
These  two  results  show  that 

<^  AHC  >  «9C  BCD. 
Thus  the  angle  HCD  must  be  acute,  as  ^  is  a  right  angle. 
It  follows  from  Prop.  XI.,  XII.  that  the  lines  AB  and  CD 
intersect.^ 

This  result  allows  Saccheri  to  conclude  that  the  Hypo- 
thesis of  the  Obtuse  Angle  is  false  [Prop.  XIV.].  In  fact,  on 
this  hypothesis  Euclid's  Postulate  holds  [Prop.  XIII.],  and 
consequently,  the  usual  theorems  which  are  deduced  from 
this  postulate  also  hold.  Thus  the  sum  of  the  angles  of  the 
fundamental  quadrilateral  is  equal  to  four  right  angles,  so 
that  the  Hypothesis  of  the  Eight  Angle  is  true.^ 

§  i6.  But  Saccheri  wishes  to  prove  that  the  Fifth 
Postulate  is  true  in  every  case.  He  thus  sets  himself  to 
destroy  the  Hypothesis  of  the  Acute  Angle. 

To  begin  with  he  shows  that  o?i  this  hypothesis,  a  straight 
line  being  given,  there  can  be  drawn  a  perpendicular  to  it  and 
a  line  cutting  it  at  an  acute  angle,  which  do  not  intersect  each 
other  [Prop.  XVII.]. 

To  construct  these  lines,  let  ^-5C  (Fig.  22)  be  a  triangle 
of  which  the  angle  C  is  a  right  angle.  At  B  draw  BD  mak- 
ing the  angle  ABD  equal  to  the  angle  BAC.   Then,  on  the 


1  This  proof  is  also  found  in  the  work  of  Nasìr-Eddìn,  which 
evidently  inspifed  the  investigations  of  Saccheri. 

2  It  should  be  noted  that  in  this  demonstration  SacCHERI 
makes  use  of  the  special  type  of  argument  of  which  we  spoke  in 
Sii.  In  fact,  from  the  assumption  that  the  Hypothesis  of  the  Ob- 
tuse Angle  is  true,  we  arrive  at  the  conclusion  that  the  Hypothesis 
of  the  Right  Angle  is  true.  This  is  a  characteristic  form  taken  in 
such  cases  by  the  ordinary  reductio  ad  absurdum  argument. 


Saccheri  and  the  Third  Hypothesis. 


39 


Hypothesis  of  the  Acute  Atigle,  the  angle  CBD  is  acute,  and 
of  the  two  hnes  CA,  BD,  which  do  not  meet  [Bk.  I,  27], 
one  makes  a  right  angle  with  BC. 

In  what  follows  we  consider  only  the  Hypothesis  of  the 
Acute  Aiigle. 

Let  (Fig.  23)  a,b  be  two  straight  lines  in  the  same  plane 
which  do  not  meet. 


A, 


A2 


^ 


Fig.  23 


From  the  points  A^,  A^^  on  a  draw  perpendiculars 
A^Bt,,  A.^B^  to  b. 

The  angles  A^,  A,  of  the  quadrilateral  thus  obtained 
can  be 

(i)  one  right,  and  one  acute: 

(ii)  both  acute: 

(iii)  one  acute  and  one  obtuse. 

In  the  first  case,  there  exists  already  a  common  per- 
pendicular to  the  two  lines  a,  b. 

In  the  second  case,  we  can  prove  the  existence  of  such 
a  common  perpendicular  by  using  the  idea  of  continuity 
[Saccheri,  Prop.  XXII.].  In  fact,  if  the  straight  line  A-,  B^  is 
moved  continuously,  while  kept  perpendicular  to  b,  until  it 
reaches  the  position  A^B^,  the  angle  B^At_A2  starts  as  an 
acute  angle  and  increases  until  it  becomes  an  obtuse  angle. 
There  must  be  an  intermediate  position  AB  in  which  the 
angle  BAA^  is  a  right  angle.  Then  AB  is  the  common 
perpendicular  to  the  two  lines  a,  b. 

In  the  third  case,  the  lines  a,  b  do  not  have  a  common 


40 


II.     The  Forerunners  of  Non-Euclidean  Geometry. 


perpendicular,  or,  if  such  exists,  it  does  not  fall  between  B^ 
and  B2. 

Evidently  there  will  be  no  such  perpendicular  if,  for  all 
the  points  Ar  situated  upon  a,  and  on  the  same  side  of  A^, 
the  quadrilateral  ^i^.^r^;-  has  always  an  obtuse  angle  at  Ar. 

With  this  hypothesis  of  the  existence  of  two  coplanar 
straight  lines  which  do  not  intersect,  and  have  no  common 
perpendicular,  Saccheri  proves  that  such  lines  always  ap- 
proach nearer  and  nearer  to  each  other  [Prop.  XXIII.],  and  that 
their  distance  apart  finally  becomes  smaller  than  any  segment, 
taken  as  small  as  we  please  [Prop.  XXV.].  In  other  words, 
if  there  are  two  coplanar  straight  Hues,  which  do  not  cut 
each  other,  and  have  no  common  perpendicular,  then  these 
lines  must  be  asymptotic  to  each  other." 

To  prove  that  such  asymptotic  lines  effectively  exist, 
Saccheri  proceeds  as  follows: — ^ 


Fig.  24. 


Among  the  lines  of  the  pencil  through  A,  coplanar  with 
the  line  b,  there  exist  lines  which  cut  b,  as,  e.  g.,  the  line 
AB  perpendicular  to  b;  and  lines  which  have   a  common 


1  With  this  result  the  question  raised  by  the  Greeks,  as  to 
the  possibility  of  asymptotic  lines  in  the  same  plane,  is  answered 
in  the  affirmative.     Cf.  p.  3. 

2  The  statement  of  Saccheri's  argument  upon  the  asymptotic 
lines  differs  in  this  edition  from  that  given  in  the  Italian  and 
German  editions.  The  changes  introduced  were  suggested  to  me 
by  some  remarks  of  Professor  Carslaw. 


The  Existence  of  Asymptotic  Lines  ai 

perpendicular  with  ò,  as,  e.  g.,  the  line  AA'  perpendicular 
to  A£  [cf.  Fig.  24]. 

If  AI'  cuts  Ò,  every  other  line  of  the  pencil,  which 
makes  a  smaller  angle  with  AB  than  the  acute  angle  BAjP, 
also  cuts  è.  On  the  other  hand^  if  the  line  A  Q,  different  from 
AA',  has  a  common  perpendicular  with  ò,  every  other  line, 
which  makes  with  AB  a  larger  acute  angle  than  the  angle 
BAQ,  has  a  common  perpendicular  with  ò  [cf  §  39, 
case  (ii).] 

Also  it  is  clear  that,  if  we  take  the  lines  of  the  pencil 
through  A,  from  the  ray  AB  towards  the  ray  AA',  we  shall 
not  find,  among  those  which  cut  d,  any  line  which  is  the  last 
line  of  that  set.  In  other  words,  the  angles  BAB,  which  the 
lines  AB,  cutting  ^,  make  with  AB,  have  an  i/J'per  limit,  the 
angle  BAX,  such  that  the  line  AX  does  not  cut  b. 

Then  Saccheri  proves  [Prop.  XXX.]  that,  if  we  start  with 
AA  and  proceed  in  the  pencil  through  A  in  the  direction 
opposite  to  that  just  taken,  we  shall  not  find  any  last  line  in 
the  set  of  lines  which  have  a  common  perpendicular  with  b\ 
that  is  to  say,  the  angles  BA  Q,  where  A  Q  has  a  common 
perpendicular  with  b,  have  a  lower  limit,  the  angle  BA  V, 
such  that  the  line  ^y  does  not  cut  b  and  has  not  a  com- 
mon perpendicular  with  b. 

It  follows  that  A  Vis  a.  line  asymptotic  to  b. 

Further  Saccheri  proves  that  the  two  hnes  AX  and  A  V 
coincide  [Prop.  XXXII.].  His  argument  depends  upon  the 
consideration  of  points  at  infinity;  and  it  is  better  to  sub- 
stitute for  it  another,  founded  on  his  Prop.  XXI.,  viz.,  On  the 
Hypothesis  of  the  Right  Angle,  and  on  that  of  the  Acute  Angle, 
the  distance  of  a  point  on  one  of  the  lines  containing  an  angle 
from  the  other  bounding  line  increases  indefinitely  as  this  point 
moves  further  and  further  along  the  line. 


42 


II.    The  Forerunners  of  Non-Euclidean  Geometry. 
The  suggested  argument  is  as  follows: 

Ar^= p 


Fig.  25- 

If  AX  [Fig.  25]  does  not  coincide  with  A  Y,  we  can  take 
a  point  P  on  AY,  such  that  the  perpendicular  FF'  from  F 
to  AX  satisfies  the  inequality 

(  1 1  FF'  >  AB.  [Prop.  XXL] 

On  the  other  hand,  if  FQ  is  the  perpendicular  from  F  to  b, 
the  property  of  asymptotic  lines  [Prop.  XXIII]  shows  that 
AB>FQ. 

But  F  is  on  the  opposite  side  of  AX  from  I?. 
Therefore  PQ  >  FF. 

Combining  this  inequality  with  the  preceding,  we  find  that 

AB>PF. 
which  contradicts  (i). 

Hence  AX  coincides  with  A  Y. 

We  may  sum  up  the  preceding  results  in  the  following 
theorem  : — 

A 


t  B  b 

Fig.  26. 

On  the  Hypothesis  of  the  Acute  Angle,  there  exist  in  the 
pencil  of  lines  through  A  two  lities  p  and  q,  asymptotic  to  b, 
one  towards  the  right,  and  the  other  towards  the  left,  which 
divide  the  pencil  into  two  parts.  The  first  of  these  consists  of 
the  lines  which  intersect  b,  and  the  second  of  those  which  have  a 
common  perpendicular  ivith  it.^ 


I  In  Saccheri's  work    tliere   will  be  found  many  other  inter- 
esting   theorems    before    he    reaches    this    result.      Of    these    the 


Saccheri's  Conclusion. 


43 


§  17.  At  this  point  Saccheri  attempts  to  come  to  a 
decision,  trusting  to  intuition  and  to  faith  in  the  validity  of 
the  Fifth  Postulate  rather  than  to  logic.  To  prove  that  the 
Hypothesis  of  the  Acute  Angle  is  absolutely  false,  because  it  is 
repugnant  to  the  7iature  of  the  straight  line  [Prop.  XXXIIL]  he 
relies  upon  five  LemmaS;,  spread  over  sixteen  pages.  In  sub- 
stance, however,  his  argument  amounts  to  the  statement 
that  if  the  Hypothesis  of  the  Acute  Angle  were  true,  the 
lines  p  (Fig.  2  6)  and  b  would  have  a  comfnon  perpendicular 
at  their  conunon  point  at  i?iftnity,  which  is  contrary  to  the 
nature  of  the  straight  lifie.  The  so-called  demonstration  of 
Saccheri  is  thus  founded  upon  the  extension  to  irifnity  of 
certain  properties  which  are  valid  for  figures  at  a  finite 
distance. 

However,  Saccheri  is  not  satisfied  with  his  reasoning 
and  attempts  to  reach  the  wished-for  proof  by  adopting 
anew  the  old  idea  of  equidistance.  It  is  not  worth  while  to 
reproduce  this  second  treatment  as  it  does  not  contain  any- 
thing of  greater  value  than  the  discussions  of  his  prede- 
cessors. 

Stillj  though  it  failed  in  its  aim,  Saccheri's  work  is  of 
great  importance.  In  it  the  most  determined  eftort  had  been 
made  on  behalf  of  the  Fifth  Postulate;  and  the  fact  that  he 
did  not  succeed  in  discovering  any  contradictions  among 
the  consequences  of  the  Hypothesis  of  the  Acute  Angle,  could 
not  help  suggesting  the  question,  whether  a  consistent  log- 
ical geometrical  system  could  not  be  built  upon  this  hypo- 


following  is  noteworthy:  If  two  straight  liiies  continually  approach 
each  other  and  their  distance  apart  remains  always  greater  than  a 
given  segment,  then  the  Hypothesis  of  the  Acute  Angle  is  impossible. 
Thus  it  follows  that,  if  we  postulate  the  absence  of  asymptotic 
straight  lines,  we  must  accept  the  truth  of  the  Euclidean  hypo- 
thesis. 


AA  TI.     The  Forerunners  of  Non-Euclidean  Geometry. 

thesis,  and  the  Euclidean  Postulate  be  impossible  of  demon- 
stration.^ 

Johann  Heinrich  Lambert  [1728 — 1777]- 
§  18.  It  is  difficult  to  say  what  influence  Saccheri's 
work  exercised  upon  the  geometers  of  the  iS^li  century. 
However,  it  is  probable  that  the  Swiss  mathematician 
Lambert  \vas  familiar  with  it,  ^  since  in  his  Theorie  der  Par- 
allellitiien  [1766]  he  quotes  a  dissertation  by  G.  S.  Klugel 
[1739 — i8i2]3,  where  the  work  of  the  Italian  geometer 
is  carefully  analysed.  Lambert's  Theorie  der  Fara/lellmien 
was  published  after  the  author's  death,  being  edited  by 
J.  Bernoulli  and  C.  F.  Hindenburg.  It  is  divided  into 
three  parts.  The  first  part  is  of  a  critical  and  philosophical 
nature.  It  deals  with  the  two-fold  question  arising  out  of  the 
Fifth  Postulate:  whether  it  can  be  proved  with  the  aid  of 
the  preceding  propositions  only,  or  whether  the  help  of  some 
other  hypothesis  is  required.    The  second  part  is  devoted  to 


1  The  publication  of  Saccheri's  work  attracted  considerable 
attention.  Mention  is  made  of  it  in  two  Histories  of  Mathematics: 
that  of  J.  C.  Heilbronner  (Leipzig,  1742)  and  that  of  Montucla 
(Paris,  1758).  Further  it  is  carefully  examined  by  G.  S.  Klugel 
in  his  dissertation  noted  below  (Note  (3)).  Nevertheless  it  was 
soon  forgotten.  Not  till  1889  did  E.  Beltrami  direct  the  attention 
of  geometers  to  it  again  in  his  Note:  Un  precursore  italiatio 
di  Legendre  e  di  Lobatschewsky.  Rend.  Ace.  Lincei  (4),  T.  V.  p.  441 
— 448.  Thereafter  Saccheri's  work  was  translated  into  English  by 
G.  B.  Halsted  (Amer.  Math.  Monthly,  Vol.  I.  1S94  et  seq.);  into 
German,  by  Engel  and  Stackel  (77/.  der  P.  1895);  into  Italian, 
by  G.  Boccardini  (Milan,  Hoepli,  1904). 

2  Cf.  SegrE:  Congetture  intorno  alla  influenza  di  Girolamo 
Saccheri  sulla  forrjiazione  della  geometria  ìion  euclidea.  Atti  Acc. 
Scienze  di  Torino,  T.  XXXVIIL  (1903). 

3  Conatiiufn  praecipuorum  theoriam  parallelarum  demonstrandi 
recensio,  guani  publico  examini  submitteni  A.  G.  Kaestner  et  auctor 
respondens  G.  S.  Kliigel,  (Gòttingen,  1763). 


Lambert's  Three  Hypotheses. 


45 


the  discussion  of  different  attempts  in  which  the  Euclidean 
Postulate  is  reduced  to  very  simple  propositions,  which 
however,  in  their  turn,  require  to  be  proved.  The  third,  and 
most  important,  part  contains  an  investigation  resembling 
that  of  Saccheri,  of  which  we  now  give  a  short  summary/ 

§  19.  Lambert's  fundamental  figure  is  a  quadrilateral 
with  three  right  angles,  and  three  hypotheses  are  made  as  to 
the  nature  of  the  fourth  angle.  The  first  is  the  Hypothesis 
of  the  Right  Angle;  the  second,  the  Hypothesis  of  the  Obtuse 
Angle;  and  the  third,  the  Hypothesis  of  the  Acute  Angle.  Also 
in  his  treatment  of  these  hypotheses  the  author  does  not 
depart  far  from  Saccheri's  method. 

^\vt  first  hypothesis  leads  easily  to  the  Euclidean  system. 

In  rejecting  the  second  hypothesis,  Lambert  relies  upon 
a  figure  formed  by  two  straight  lines  a,  b,  perpendicular  to 
a  third  line  ^^  (Fig.  27).  From  points  £,  B^,  B^y.-Bn, 
taken  in  succession  upon    B       Bj    B,  B„ 

the  line  b,  the  perpen- 
diculars, BAy  B-,A^,  B^A^, 
:  .  B„An  are  drawn  to  the 
hne  a.   He  proves,  in  the 

first  place,  that  these  per-     A       A.^    Aj  An 

pendiculars       continually  ^'^-  ^7- 

diminish,  starting  from  the  perpendicular  BA.  Next,  that 
the  difterence  between  each  and  the  one  which  succeeds  it 
continually  increases. 

Therefore  we  have 

BA—BnAn  >  n  {BA—B^A^. 

But,  if  n  is  taken  sufficiently  large,  the  second  member 


I  Cf.  Magazin  fur  reine  und  angewandte  Math.,  2.  Stuck, 
p.  137 — 164.  3.  Stuck,  p.  325 — 358,  (1786).  Lambert's  work  was 
again  published  by  Engel  and  Stackel  {Th.  der  P.)  p.  135 — 208, 
preceded  by  historical  notes  on  the  author. 


aF)  li.     The  Forerunners  of  Non-Euclidean  Geometry. 

of  this  inequality  becomes  as  great  as  we  please  {Postulate 
of  Archimedes]  \  whilst  the  first  member  is  always  less  than 
£A.  This  contradiction  allows  Lambert  to  declare  that  the 
second  hypothesis  is  false. 

In  examining  the  third  hypothesis,  Lambert  again  avails 
himself  of  the  preceding  figure.  He  proves  that  the  perpen- 
diculars £A,  BxA^,  .  .  B,iAn  continually  increase,  and  that 
at  the  same  time  the  difference  between  each  and  the  one 
which  precedes  it  continually  increases.  As  this  result  does 
not  lead  to  contradictions,  like  Saccheri  he  is  compelled  to 
carry  his  argument  further.  Then  he  finds,  that,  on  the  third 
hypothesis  the  sum  of  the  angles  of  a  triangle  is  less  than 
two  right  angles;  and  going  a  step  further  than  Saccheri, 
he  discovers  that  the  defect  of  a  polygon,  that  is,  the  differ- 
ence between  2  {n — 2)  right  angles  and  the  sum  of  its  angles, 
is  proportional  to  the  area  of  the  polygon.  This  result  can 
be  obtained  more  easily  by  observing  that  both  the  area  and 
the  defect  of  a  polygon,  which  is  the  sum  of  several  others, 
are,  respectively,  the  sum  of  the  areas  and  of  the  defects  of 
the  polygons  of  which  it  is  composed.^ 

§  20.  Another  remarkable  discovery  made  by  Lambert 
has  reference  to  the  measurement  of  geometrical  magnitudes. 
It  consists  precisely  in  this,  that,  whilst  in  the  ordinary  geo- 
metry only  a  relative  meaning  attaches  to  the  choice  of  a 


1  The  Postulate  of  Archimedes  is  again  used  here  in  a  form 
which  assumes  the  infinity  of  the  straight  line  (cf.  Saccheri,  Note 
P-  37)- 

2  It  is  right  to  point  out  that  in  the  Hypothesis  af  the  Aade 
Angle  Saccheri  had  already  met  the  defect  here  referred  to,  and 
also  noted  implicitly  that  a  quadrilateral,  made  up  of  several 
others,  has  for  its  defect  the  sum  of  the  defects  of  its  parts  (Prop. 
XXV).  However  he  did  not  draw  any  conclusion  from  this  as  to 
the  area  being  proportional  to  the  defect. 


Relative  and  Absolute  Units. 


47 


particular  unit  in  the  measurement  of  lines,  in  the  geometry 
founded  upon  the  third  hypothesis^  we  can  attach  to  it  an 
absolute  meaning. 

First  of  all  we  must  explain  the  distinction,  which  is 
here  introduced,  between  absolute  and  relative.  In  many 
questions  it  happens  that  the  elements,  supposed  given,  can 
be  divided  into  two  groups,  so  that  those  oi  i\\Q  first  grotip 
remain  fixed,  right  through  the  argument,  while  those  of  the 
second  group  may  vary  in  a  number  of  possible  cases.  When 
this  happens,  the  explicit  reference  to  the  data  of  the  first 
group  is  often  omitted.  All  that  depends  upon  the  varying 
data  is  considered  relative;  all  that  depends  upon  the  fixed 
data  is  absolute. 

For  example,  in  the  theory  of  the  Domain,  of  Ration- 
ality, the  data  of  the  second  group  [the  variable  data]  are 
taken  as  certain  simple  irrationalities  [constituting  a  base\., 
and  "ùx^  first  group  consists  simply  of  unity  [i],  which  is 
often  passed  over  in  silence  as  it  is  common  to  all  domains. 
In  speaking  of  a  number,  we  say  that  it  is  rational  relatively 
to  a  given  base,  if  it  belongs  to  the  domain  of  rationality 
defined  by  that  base.  We  say  that  it  is  rational  absolutely, 
if  it  is  proved  to  be  rational  with  respect  to  the  base  i, 
which  is  common  to  all  domains. 

Passing  to  Geometry,  we  observe  that  in  every  actual 
problem,  we  generally  take  certain  figures  as  given  and 
therefore  the  magnitudes  of  their  parts.  In  addition  to  these 
variable  data  [of  the  second group\  which  can  be  chosen  in 
an  arbitrary  manner,  there  is  always  implicitly  assumed  the 
presence  of  the  fundamental  figures,  straight  lines,  planes, 
pencils,  etc.  [fixed  data  or  of  the  first  group].  Thus,  every 
construction,  every  measurement,  every  property  of  any 
figure  ought  to  be  held  as  relative,  if  it  is  essentially  relative 
to  the  variable  data.  It  ought,  on  the  other  hand,  to  be 
spoken  of  as  absolute,  if  it  is  relative  only  to  the  fixed  data 


^8  II-     The  Forerunners  of  Non-Euclidean  Geometry. 

[the  fundamental  figures],  or,  if,  being  enunciated  in  terms 
of  the  variable  data,  it  only  appears  to  depend  upon  them, 
so  that  it  remains  fixed  when  these  vary. 

In  this  sense  it  is  clear  that  in  ordinary  geometry  the 
measurement  of  lines  has  necessarily  a  relative  meaning. 
Indeed  the  existence  of  similar  figures  does  not  allow  us  in 
any  way  to  individualize  the  size  of  a  line  in  terms  of  funda- 
mental figures  [straight  line,  pencil,  etc.]. 

For  an  angle  on  the  other  hand,  we  can  choose  a  method 
of  measurement  which  expresses  one  of  its  absolute  pro- 
perties. It  is  sufficient  to  take  its  ratio  to  the  angle  of  a 
complete  revolution,  that  is,  to  the  entire  pencil,  this  being 
one  of  the  fundamental  figures. 

We  return  now  to  Lambert  and  his  geometry  corre- 
sponding to  the  third  hypothesis.  He  observed  that  with 
every  segment  we  can  associate  a  definite  angle,  which  can 
easily  be  constructed.  From  this  it  follows  that  every  seg- 
ment is  brought  into  correspondence  with  the  fundamental 
figure  [the  pencil].  Therefore,  in  the  new  [hypothetical] 
geometry,  we  are  entitled  to  ascribe  an  absolute  meaning 
also  to  the  measurement  of  segments. 

To  show  in  the  simplest  way  how  to  every  segment  we 
can  find  a  corresponding  angle,  and  thus  obtain  an  ab- 
solute numerical  measurement  of  lines,  let  us  imagine  an 
equilateral  triangle  constructed  upon  every  segment.  We 
are  able  to  associate  with  every  segment  the  angle  of  the 
triangle  corresponding  to  it  and  then  the  measure  of  this 
angle.  Thus  there  exists  a  one-one  correspondence  between 
segments  and  the  angles  comprised  between  certain  limits. 

But  the  numerical  representation  of  segments  thus  ob- 
tained does  not  enjoy  the  distributive  property  which  belongs 
to  lengths.  On  taking  the  sum  of  two  segments,  we  do  not 
obtain  the  sum  of  the  corresponding  angles.  However,  a 
function  of  the  angle,  possessing  this  property,  can  be  ob- 


The  Absolute  Unit  of  Length.  aq 

tained,  and  we  can  associate  with  the  segment,  not  the  said 
angle,  but  this  function  of  the  angle.  For  every  value  of  the 
angle  between  certain  limits,  such  a  function  gives  an  absolute 
vieasure  of  segments.  The  absolute  unit  of  length  is  that 
segment  for  which  this  function  takes  the  value  i. 

Now  if  a  certain  function  of  the  angle  is  distributive  in 
the  sense  just  indicated,  the  product  of  this  function  and  an 
arbitrary  constant  also  possesses  that  property.  It  is  there- 
fore clear  that  we  can  always  choose  this  constant  so  that 
the  absolute  unit  segment  shall  be  that  segment  which  corre- 
sponds to  any  assigned  angle:  e.  g.,  45".  The  possibility  of 
constructing  the  absolute  unit  segment,  given  the  angle,  de- 
pends upon  the  solution  of  the  following  problem  : 

To  construct,  on  the  Hypothesis  of  the  Acute  Angle,  an 
equilateral  triangle  with  a  given  defect. 

So  far  as  regards  the  absolute  m.easure  of  the  areas  of 
polygons,  we  remark  that  it  is  given  at  once  by  the  defect 
of  the  polygons.  We  can  also  assign  an  absolute  measure 
for  polyhedrons. 

But  with  our  intuition  of  space  the  absolute  measure 
of  all  these  geometrical  magnitudes  seems  to  us  impossible. 
Hence  if  tue  deny  the  existence  of  an  absolute  unit  for  segments, 
we  can,  with  Lambert,  reject  the  third  hypothesis. 

§  21.  As  Lambert  realized  the  arbitrary  nature  of  this 
statement,  let  it  not  be  supposed  that  he  believed  that  he 
had  in  this  way  proved  the  Fifth  Postulate. 

To  obtain  the  wished-for  proof,  he  proceeds  with  his 
investigation  of  the  consequences  of  the  third  hypothesis,  but 
he  only  succeeds  in  transforming  his  question  into  others 
equally  difficult  to  answer. 

Other  very  interesting  points  are  contained  in  the 
Theorie  der  Parallellinien,  for  example,  the  close  resemblance 

4 


co  II.     The  Forerunners  of  Non-Euclidean  Geometry. 

to  spherical  geometry^  of  the  plane  geometry  which  would 
hold,  if  the  second  hypothesis  were  valid,  and  the  remark  that 
spherical  geometry  is  independent  of  the  Parallel  Postulate, 
Further^  referring  to  the  third  hypothesis^  he  made  the  follow- 
ing acute  and  original  observation:  Froin  this  I  should  al- 
most conchcde  that  the  third  hypothesis  tvould  occur  in  the  case 
of  an  imaginary  sphere. 

He  was  perhaps  brought  to  this  way  of  looking  at  the 
question  by  the  formula  {A-\-B-\-  C — it)  r^,  which  expresses 
the  area  of  a  spherical  triangle.  If  in  this  we  write  for  the 
radius  r,  the  imaginary  radius  K -i  r  we  obtain 

r^\yi—A—B—C\; 
that  is,  the  formula  for  the  area  of  a  plane  triangle  on 
Lambert's  third  hypothesise 

§  22.  Lambert  thus  left  the  question  in  suspense.  In- 
deed the  fact  that  he  did  not  publish  his  investigation  allows 
us  to  conjecture  that  he  may  have  discovered  another  way 
of  regarding  the  subject. 

Further,  ,it  should  be  remarked  that,  from  the  general 
want  of  success  of  these  attempts,  the  conviction  began  to 
be  formed  in  the  second  half  of  the  a  8th  Century  that  it 
would  be  necessary  to  admit  the  Euclidean  Postulate^  or 
some  other  equivalent  postulate,  without  proof. 

In  Germany,  where  the  writings  upon  the  question 
followed  closely  upon  each  other,  this  conviction  had  al- 
ready assumed  a  fairly  definite  form.  We  recognize  it  in 
A,  G.  Kastner,^  a  well-known  student  of  the  theory  of 
parallels,  and  in  his   pupil,   G.  S.  Klugel,  author   of  the 


1  In  fact,  in  Spherical  Geometry  the  sum  of  the  angles  of  a 
quadrilateral  is  greater  than  four  right  angles,  etc. 

2  Cf.  Engel  u.  Stackel;  Th.  der  P.  p.  146. 

3  For  some  information  about  Kastner,  cf.  Engel  u.  StAckel; 
Th.  der  P.  p.    139 — 141. 


Klùgel's  Work.  e  i 

valuable  criticism  of  the  most  celebrated  attempts  to  de- 
monstrate the  Fifth  Postulate,  referred  to  on  p.  44  [note  3]. 
In  this  work  Klugel  finds  each  of  the  proposed  proofs 
insufficient  and  suggests  the  possibility  of  non-intersecting 
straight  lines  being  divergent  YMoglich  ware  es  freilick,  da^ 
Gerade,  die  sich  nihct  schneiden,  voiieinander  abweiche?i\.  He 
adds  that  the  apparent  contradiction  which  this  presents  is 
not  the  result  of  a  rigorous  proof,  nor  a  consequence  of  the 
definitions  of  straight  lines  and  curves,  but  rather  something 
derived  from  experience  and  the  judgment  of  our  senses. 
\Dafi  so  etwas  widersinnig  ist,  wissen  wir  nicht  infolge  strenger 
Sc/iiusse  Oder  vcrmoge  deutlicher  Begriffe  V07i  der  geraden  und 
der  kntmmen  Linie,  viebnehr  durch  die  Erfahrung  und  durch 
das  Urteil  unserer  Augen]. 

The  investigations  of  Saccheri  and  Lambert  tend  to 
confirm  Klugel's  opinion,  but  they  cannot  be  held  to  be 
a  proof  of  the  impossibihty  of  demonstrating  the  Euclidean 
hypothesis.  Neither  would  a  proof  be  reached  if  we  proceed- 
ed along  the  way  opened  by  these  two  geometers,  and  de- 
duced any  number  of  other  propositions,  not  contradicting 
the  fundamendal  theorems  of  geometry. 

Nevertheless  that  one  should  go  forward  on  this  path, 
without  Saccheri's  presupposition  that  contradictions  would 
be  found  there,  constitutes  historically  the  decisive  step  in  the 
discovery  that  Euclìd's  Postulate  could  not  be  proved,  and 
in  the  creation  of  the  Non-Euclidean  geometries. 

But  from  the  work  of  Saccheri  and  Lambert  to  that  of 
LoBATSCHEWSKY  and  B0LYAI,  which  is  based  upon  the  above 
idea,  more  than  half  a  century  had  still  to  pass  ! 

The  French  Geometers  tov,;'ards  the  End  of  the 
i8th  Century. 
§  23.     The  critical  study  of  the  theory  of  parallels, 
which  had  already  led  to  results  of  great  interest  in  Italy  and 


(-2  II.     The  Forerunners  of  Non-Euclidean  Geometry. 

Germany,  also  made  a  remarkable  advance  in  France  to- 
wards the  end  of  the  iSth  Century  and  the  beginning  of 
the  19th. 

D'Alembert  [1717  — 1783];  in  one  of  his  articles  on 
geometry,  states  that  'La  definition  et  les  propriétés  de  la 
ligne  droite,  ainsi  que  des  lignes  parallèles  sont  l'écueil  et 
pour  ainsi  dire  le  scandale  des  elements  de  Geometrie.'  ^ 
He  holds  that  with  a  good  definition  of  the  straight  line 
both  difficulties  ought  to  be  avoided.  He  proposes  to  define 
a  parallel  to  a  given  straight  line  as  any  other  coplanar 
straight  line,  which  joins  two  points  which  are  on  the  same 
side  of  and  equally  distant  from  the  given  line.  This  definition 
allows  parallel  lines  to  be  constructed  immediately.  However 
it  would  still  be  necessary  to  show  that  these  parallels  are 
equidistant.  This  theorem  was  offered,  almost  as  a  challenge, 
by  D'Alembert  to  his  contemporaries. 

§  24.  De  Morgan,  in  his  Budget  of  Paradoxes^,  relates 
that  Lagrange  [1736 — 1813],  towards  the  end  of  his  life, 
wrote  a  memoir  on  parallels.  Having  presented  it  to  the 
French  Academy,  he  broke  off"  his  reading  of  it  with  the  ex- 
clamation: 'II  faut  que  j'y  songe  encore!'  and  he  withdrew 
the  MSS. 

Further  Houel  states  that  Lagrange,  in  conversation 
with  BiOT,  affirmed  the  independence  of  Spherical  Trigon- 
ometry from  Euclid's  Postulate.-^  In  confirmation  of  this 
statement  it  should  be  added  that  Lagrange  had  made  a  spe- 
cial study  of  Spherical  Trigonometry,'^  and  that  he  inspired. 


1  Cf.  D'Alembert:  Melanges  de  Littcrature,  d'Hisioire,  et 
de  Philosophie,  T.  V.  S  II  (l7S9)-  Also:  Encychfédie  Méihodiqiie 
Mathématique ;  T.  II.  p.  519,  Article:  Parallèles  (1785). 

2  A.  DE  Morgan:  A  Budget  of  Paiadoxes,^.\'J2,.  (London,  1872). 

3  Cf.  J.  Houel:  Essai  critique  sur  les  principes  fondamenlaux 
de  la  geometrie  élèmentaire,    p.  84,    Note    (Paris,  G.  VJLLARS,   1 883). 

4  Cf.  Miscellanea  Taurinensia,  T.  II.  p.  299—322  (1760 — 5i). 


D'Alembert,  Lagrange,  and  Laplace.  £2 

if  he  did  not  write,  a  memoir  ''Sur  les  principes  fondamentaux 
de  la  Mecanique  [1760 — i]^,  in  which  Foncenex  discussed 
a  question  of  independence,  analogous  to  that  above  noted 
for  Spherical  Trigonometry.  In  fact,  Foncenex  shows  that 
the  analytical  law  of  the  Composition  of  Forces  acting  at  a 
point  does  not  depend  on  the  Fifth  Postulate,  nor  upon  any 
other  which  is  equivalent  to  it.^ 

§  25.  The  principle  of  similarity,  as  a  fundamental 
notion,  had  been  already  employed  by  Wallis  in  1663  [cf. 
§  9].  It  reappears  at  the  beginning  of  the  19th  Century,  sup- 
ported by  the  authority  of  two  famous  geometers:  L.  N.  M. 
Carxot  [1753 — 1823]  and  Laplace  [1749 — 1827]. 

In  a  Note  [p.  481]  to  his  Geometrie  de  Position  [1803] 
Carnot  affirms  that  the  theory  of  parallels  is  allied  to  the 
principle  of  similarity,  the  evidence  for  which  is  almost  on 
the  same  plane  as  that  for  equality,  and  that,  if  this  idea  is 
once  admitted,  it  is  easy  to  establish  the  said  theory  rigorously. 

Laplace  [1824]  observes  that  Newton's  Law  [the  Law 
of  Gravitation],  by  its  simplicity,  by  its  generality  and  by  the 
confirmation  which  it  finds  in  the  phenomena  of  nature,  must 
be  regarded  as  rigorous.  He  then  points  out  that  one  of  its 
most  remarkable  properties  is  that,  if  the  dimensions  of  all 
the  bodies  of  the  universe,  their  distances  from  each  other, 
and  their  velocities,  were  to  decrease  proportionally,  the 
heavenly  bodies  would  describe  curves  exactly  similar  to 
those  which  they  now  describe,  so  that  the  universe,  reduced 
step  by  step  to  the  smallest  imaginable  space,  would  always 
present  the  same  phenomena  to  its  observers.  These  pheno- 
mena, he  continues,  are  independent  of  the  dimensions  of  the 
universe,  so  that  the  simphcity  of  the  laws  of  nature  only  allows 
the  observer  to  recognise  their  ratios.   Referring  again  to  this 

1  Cf.  Lagrange:   Oeiivres,  T.  VIL  p.  331 — 2fil- 

2  Cf.  Chapter  VL 


Ca  II.     The  Forerunners  of  Non-Euclidean  Geometry. 

astronomical  conception  of  space,  he  adds  in  a  Note:  'The 
attempts  of  geometers  to  prove  Euclid's  Postulate  on  Parallels 
have  been  up  till  now  futile.  However  no  one  can  doubt  this 
postulate  and  the  theorems  which  Euclid  deduced  from  it.  Thus 
the  notion  of  space  includes  a  special  property,  self-evident, 
without  which  the  properties  of  parallels  cannot  be  rigorously 
established.  The  idea  of  a  bounded  region,  e.  g.,  the  circle, 
contains  nothing  which  depends  on  its  absolute  magnitude. 
But  if  we  imagine  its  radius  to  diminish,  we  are  brought 
without  fail  to  the  diminution  in  the  same  ratio  of  its  circum- 
ference and  the  sides  of  all  the  inscribed  figures.  This  pro- 
portionality appears  to  me  a  more  natural  postulate  than 
that  of  Euclid,  and  it  is  worthy  of  note  that  it  is  discovered 
afresh  in  the  results  of  the  theory  of  universal  gravitation.'  ^ 

§  26.  Along  with  the  preceding  geometers,  it  is  right 
also  to  mention  J.  B.  Fourier  [1768 — 1830],  for  a  discussion 
on  the  straight  line  which  he  carried  on  with  Monge.^  To 
bring  this  discussion  into  line  with"  the  investigations  on 
parallels,  we  need  only  go  back  to  D'Alembert's  idea  that 
the  demonstration  of  the  postulate  can  be  connected  with 
the  definition  of  the  straight  line  [cf  §  23]. 

Fourier,  who  regarded  the  distance  between  two  points 
as  a  prime  notion^  proposed  to  define  first  the  sphere;  then 
the  plane,  as  the  locus  of  points  equidistant  from  two 
given  points;^  then  the  straight  line,  as  the  locus  of  the 
points    equidistant    from   three   given   points.     This  method 


1  Cf.  Laplace.      Oeuvres,  T.  VI.  Livre,  V.  Ch.  V.  p.  472. 

2  Cf.  Seances  de  P  Ecole  ftormale:  De  bats,  T.  I.  p.  28 — ^^ 
(1795).  This  discussion  was  reprinted  in  Mathésis.  T.  IX.  p.  139 
-141   (1883)- 

3  This  definition  of  the  plane  was  given  by  Leibnitz  about 
a  century  before.  Cf.  Opuscules  et  fragtnents  incdiis,  edited  by 
L.  CouTURAT,  p.  554 — 5.     (Paris,  Alcan,  1903). 


Fourier  and  Lesfendre. 


55 


of  presenting  the  problem  of  the  foundations  of  geometry 
agrees  with  the  opinions  adopted  at  a  later  date  by  other 
geometers,  who  made  a  special  study  of  the  question  of 
parallels  [W.  Bolyai,  N.  Lobatschewsky,  de  Tilly].  In 
this  sense  the  discussion  between  Fourier  and  Monge  finds 
a  place  among  the  earliest  documents  which  refer  to  NoJi- 
Euclidea7i  geometry} 

Adrien  Marie  Legendre  [1752 — 1833I. 

§  27.  The  preceding  geometers  confined  themselves  to 
pointing  out  difficulties  and  to  stating  their  opinions  upon 
the  Postulate.  Legendre,  on  the  other  hand,  attempted  to 
transform  it  into  a  theorem.  His  "investigations,  scattered 
among  the  different  editions  of  his  Elements  de  Geometrie 
[1794 — 1823],  are  brought  together  in  his  Reflexions  sur 
différentes  manières  de  démontrer  la  théorie  des  paralleles  ou 
le  [the'orème  sur  la  somme  des  trois  afigles  du  triangle.  [Mém. 
Ac.  Se,  Paris,  T.  XIII.  1833.] 

In  jthe  most  interesting  of  his  attempts,  Legendre,  like 
Saccheri,  approaches  the  question  from  the  side  of  the  sum 
of  the  angles  of  a  triangle,  which  sum  he  wishes  to  prove 
equal  to  two  right  angles. 

With  this  end  in  view,  at  the  commencement  of  his  work 
he  succeeds  in!  rejecting  Saccheri's  Hypothesis  of  the  Obtuse 
Angle,  since  he  estabhshes  that  the  sum  of  the  angles  of  any 
triangle  is  either  less  than  ^Hypothesis  of  the  Acute  Angle]  or 
equal  to  {Hypothesis  of  the  Right  Angle]  two  right  angles. 

We  reproduce  a  neat  and  simple  proof  which  he  gives 
of  this  theorem  : 

Let  n  equal  segments  ^1^2,  -^2^3,  • . .  ^«^«+1  be  taken 


I  To  this  we  add  that  later  memoirs  and  investigations 
showed  that  Fourier's  definition  also  fails  to  build  up  the  Eucli- 
dean theory  of  parallels,  without  the  help  of  the  Fifth  Postulate, 
or  some  other  equivalent  to  it. 


c5  II.     The  Forerunners  of  Non-Euclidean  Geometry. 

one  after  the  other  on  a  straight  Hne  [Fig.  28].  On  the  same 
side  of  the  Hne  let  n  equal  triangles  be  constructed,  having 
for  their  third  angular  points  B^B^. . .  .B,f  The  segments 
Bj_B2i  BiB^,... Bn—x Bni  which  join  these  vertices,  are  equal 
and  can  be  taken  as  the  bases  of  n  equal  triangles,  B^A2B2, 

B,  E^  B3  B^  B^^2^J^3'--- -^«-i 

A,iB„.  The  figure 

is    completed   by 

adding  the  triangle 

which  is  equal  to 
the  others. 

Let  the  angle  ^i  of  the  triangle  A^B^Az  be  denoted  by 
P,  and  the  angle  A2  of  the  consecutive  triangle  by  a. 
Then  p  <  a. 

In  fact,  if  P  ^  a,  by  comparing  the  two  triangles  A^B^Az 
and  B1A2B2,  which  have  two  equal  sides,  we  would  deduce 
A,A2>B,B2. 
Further,  since  the  broken  line  A^B^Bz  . . .  -^«+1  ^«-f  i 
is  greater  than  the  segment  AiA^-^i , 

A^Bx  +  n.  B^B^  +  ^„+i  ^«+1  >  n.  A^A^, 
i.  e.,  2  Aj,Bi^n{AiA2 — B^B^). 
But  if  n  is  taken  sufficiently  great,  this  inequality  con- 
tradicts the  Postulate  of  Archimedes. 

Therefore      A^A^  is  not  greater  than  B^Bz , 
and  it  follows  that  it  is  impossible  that  P  i>>  a. 
Thus  we  have  P  <  a. 

From  this  it  readily  follows  that  the  sum  of  the  angles  ot 
the  triangle  A^B^A^  is  less  than  or  equal  to  two  right  angles. 
This  theorem  is  usually,  but  mistakenly,  called  Legendre's 
First  Theorem.  We  say  mistakenly,  because  Saccheri  had 
already  established  this  theorem  almost  a  century  earlier  [cf 
p.  38]  when  he  proved  that  the  Hypothesis  of  the  Obtuse 
Angle  was  false. 


Lea:endre's  First  Proof. 


57 


The  theorem  usually  called  Legendre's  Second  Theorem 
was  also  given  by  Saccheri,  and  in  a  more  general  form 
[cf.  p.  29].   It  is  as  follows: 

If  the  sum  of  the  angles  of  a  triangle  is  less  than  or 
equal  to  two  right  afigles  in  only  one  triangle,  it  is  respectively 
less  than  or  equal  to  two  right  angles  in  every  other  triangle. 

We  do  not  repeat  the  demonstration  of  this  theorem,  as 
it  does  not  differ  materially  from  that  of  Saccheri. 

We  shall  rather  show  how  Legendre  proves  that  the 
sum  of  the  three  angles  of  a  tria?igle  is  equal  to  two  right 
angles. 

Suppose  that  in  the  triangle  ABC  [cf.  Fig.  2  9] 
^A  ■\-  <iB  -\r  <fi  C<C  2  right  angles. 

A  point  D  being  taken  on  AB,  the  transversal  DE  is 
drawn,    making    the    angle    ADE 
equal  to  the  angle  B.   In  the  quadri- 
lateral DBCE  the  sum  of  the  angles 
is  less  than  4  right  angles. 

Therefore  ^AED^^ACB. 
The  angle  E  of  the  triangle  ADE 
is  then  a  perfectly  definite  [decreas- 
ing] function  of  the  side  AD:  or, 
what  amounts  to  the  same  thing,  the 
length  of  the  side  AD  is  fully  determined  when  we  know  the 
size  (in  right  angles)  of  the  angle  E,  and  of  the  two  fixed 
angles  A,  B. 

But  this  result  Legendre  holds  to  be  absurd,  since  the 
length  of  a  line  has  not  a  meaning,  unless  one  knows  the  unit 
of  length  to  which  it  is  referred,  and  the  nature  of  the  question 
does  not  indicate  this  unit  in  any  way. 

In  this  way  the  hypothesis 

<^A-^  ^B  +  -^  C<  2  right  angles 
is  rejected,  and  consequently  we  have 

<^A  +  <f:B  +  ^C=2  right  angles. 


58 


II.     The  Forerunners  of  Non-Euclidean  Geometry, 


Also  from  this  equality  the  proof  of  Euclid's  Postulate 
follows  easily. 

Legendre's  method  is  thus  based  upon  Lambert's  postu- 
late, which  denies  the  existence  of  an  absolute  7init  segment. 

§  28.  In  another  demonstration  Legendre  makes  use  of 
the  hypothesis: 

From  any  point  whatever,  taken  within  an  angle,  we  can 
always  draw  a  straight  line  which  7vill  cut  the  two  arms  of 
the  angled 

He  proceeds  as  follows: 

Let  ABC  he  a  triangle,  in  which,  if  possible,  the  sum  of 
the  angles  is  less  than  two  right  angles. 

Let  2  right  angles—  <^  A—^£—  <^  C=  a  [the  defect]. 
Find  the  point  A',  symmetrical  to  A,  with  respect  to  the 
side  BC.    [cf.  Fig.  30.] 

The  defect  of  the  new  tri- 
angle BCA'  is  also  a.  In  virtue 
of  the  hypothesis  enunciated 
above,  draw  through  A'  a 
transversal  meeting  the  arms 
of  the  angle  A  in  Bj^  and  C^. 
It  can  easily  be  shown  that  the 
defect  of  the  triangle  AB^  C^  is 
the  sum  of  the  defects  of  the 
four  triangles  of  which  it  is 
composed,    [cf.  also  Lambert  p.  46.] 

Thus  this  defect  is  greater  than  2  a. 
Starting  now  with  the  triangle  ABiQ  and  repeating  the 
same  construction,  we  get  a  new  triangle  whose  defect  is 
greater  than  4  a. 


Fis 


I  J.  F.  Lorenz  had  already  used  this  hypothesis  for  the  same 
purpose.  Cf.  GnaidnjS  der  reinoi  unci  angewandlen  Mathcmatik, 
(Helmstedt,  1791). 


Lesjendre's  Second  Proof. 


59 


After  n  operations  of  this  kind  a  triangle  will  have  been 
constructed  whose  defect  is  greater  than  2"  a. 

But  for  n  sufficiently  great,  this  defect,  2"  a,  must  be 
greater  than  2  right  angles  [Postulate  of  Archimedes],  which 
is  absurd. 

It  follows  that  (X  =  o,  and  ■^A-^^B-^^C=2 
right  angles. 

This  demonstration  is  founded  upon  the  Postulate  of 
Archimedes.  We  shall  now  show  how  we  could  avoid  using 
this  postulate  [cf.  Fig.  31]. 

Let  AB  and  HK  be  two  straight  lines,  of  which  AB 
makes  an  acute  angle,  and  HK  a  right  angle,  with  AH. 


Fig.  31- 


Draw  the  straight  line  AB'  symmetrical  to  AB  with  re- 
gard to  AH.  Through  the  point  H  there  passes,  in  virtue  of 
Legendre's  hypothesis,  a  line  r  which  cuts  the  two  arms  of 
the  angle  BAB' .  If  this  line  is  different  from  HK^  then  also 
the  line  /,  symmetrical  to  it  with  respect  to  AH,  enjoys  the 
same  property  of  intersecting  the  arms  of  the  angle.  It  fol- 
lows that  the  line  HK  also  meets  them. 

Thus  the  line  perpendicular  to  AH  and  a  line  making 
an  acute  angle  with  AH  always  meet. 

From  this  result  the  ordinary  theory  of  parallels  follows, 
and  <5C^  +  <^^+  ^C=  2  right  angles. 

In  other  demonstrations  Legendre  adopts  the  methods 
of  analysis  and  also  makes  an  erroneous  use  of  infinity. 


6o  n.     The  Forerunners  of  Non-Euclidean  Geometry. 

By  these  very  varied  investigations  Legendre  believed 
that  he  had  finally  removed  the  serious  difficulties  surrounding 
the  foundations  of  geometry.  In  substance,  however,  he 
added  nothing  new  to  the  material  and  to  the  results  ob- 
tained by  his  predecessors.  His  greatest  merit  lies  in  the 
elegant  and  simple  form  which  he  was  able  to  give  to  all  his 
writings.  For  this  reason  they  gained  a  wide  circle  of  readers 
and  helped  greatly  to  increase  the  number  of  disciples  of  the 
new  ideas,  which  at  that  time  were  beginning  to  be  formed. 

Wolfgang  Bolyai  [1775  —  1856]. 

§  29.  In  this  article  we  come  to  the  work  of  the  Hungarian 
geometer  W.  Bolvai.  His  interest  in  the  theory  of  parallels 
dates  back  to  the  time  when  he  was  a  student  at  Gottingen 
[1796 — 99],  and  is  probably  due  to  the  advice  of  Kastner 
and  of  his  friend,  the  young  Professor  of  Astronomy,  K.  F. 
Seyffer  [1762 — 1822]. 

In  1804  he  sent  Gauss,  formerly  one  of  his  student 
friends  at  Gottingen,  a  Theoria  Parallelarum,  which  contained 
an  attempt  at  a  proof  of  the  existence  of  equidistant  straight 
lines.^  Gauss  showed  that  this  proof  was  fallacious.  Bolvai 
however,  did  not  on  this  account  give  up  his  study  of  Axiom 
XL,  though  he  only  succeeded  in  substituting  for  it  others, 
more  or  less  evident.  In  this  way  he  came  to  doubt  the  possib- 
ility of  a  demonstration  and  to  conceive  the  impossibility 
of  doing  away  with  the  Euclidean  hypothesis.  He  asserted 
that  the  results  derived  from  the  denial  of  Axiom  XI 
could  not  contradict  the  principles  of  geometry,  since  the 
law   of  the  intersection   of  two  straight  lines,  in  its  usual 


I  The  Theoria  Parallelarum  was  written  in  Latin.  A  German 
translation  by  Engel  and  StAckel  appears  in  Math.  Ann.  Bd. 
XLIX.  p.  168—205  (1897). 


\V.  Bolyai's  Postulate. 


6i 


form,  represents  a  new  datum,  independent  of  those  which 
precede  it.^ 

Wolfgang  brought  together  his  writings  on  the  principles 
of  mathematics  in  tlie  work:  Tentamen  juventutem  studiosam 
in  elementa  Matheseos  [1832 — 33];  and  in  particular  his  in- 
vestigations on  Axiom  XI.,  while  in  each  attempt  he  pointed 
out  the  new  hypothesis  necessary  to  render  the  demon- 
stration rigorous. 

A  remarkable  postulate  to  which  Wolfgang  reduces 
Euclid's  is  the  following: 

Four  povits,  not  on  a  plane,  always  lie  2ip07i  a  sphere; 
or,  what  amounts  to  the  same  thing:  A  circle  can  always  be 
dratvn  through  three  points  not  on  a  straight  lifie.^ 

The  Euclidean  Postulate  can  be  deduced  from  this  as 
follows  [cf.  Fig.  32]: 

Let  AA,  BB'  be  two  straight  lines,  one  of  them  being 
perpendicular  to  AB.,  and  the  other  inclined  to  it  at  an  acute 
angle. 

If  we  take  a  point  M  on  the  seg- 
ment AB  between  A  and  ^,  and 
the  points  M' M".  symmetrical  to  M 
with  respect  to  the  lines  BB'  and 
AA ,  we  obtain  two  points  M' ,  M" 
not  in  the  same  straight  line  with  M. 
These  three  points  M,  M\  M"  lie 
on  the  circumference  of  a  circle.  Also 
the  lines  AA ,  BB'  must  intersect, 
since  they  both  pass  through  the  cen-  Fig.  32. 

tre  of  this  circle. 

But  from  the  fact  that  a  line  which  is  perpendicular  to 


1  Cf.  StackeL:    Die  Enideckiing  der  nichteuklidischen  Geometrie 
diirch  y.  Bolyai,  Math.  u.  Naturw.  Ber.  aus  Ungarn,  Bd.  XVII.  (1901). 

2  Cf.   W.    BoLYAi:    Kurzer  Grundriss  eines  Vermchs  etc.,  p.  46. 
(Maros  Vàsarhely,  '85  r). 


52  !!•     The  Forerunners  of  Non-Euclidean  Geometry. 

another  straight  line  and  a  line  which  cuts  it  at  an  acute  angle 
intersect,  it  follows  immediately  that  there  can  be  only  one 
parallel. 

Friedrich  Ludwig  Wachter  [1792 — 1817]. 

§  30.  When  it  had  been  seen  that  the  Euclidean  Postulate 
depends  on  the  possibiHty  of  a  circle  being  drawn  through 
any  three  points  not  on  a  straight  line,  the  idea  at  once  sug- 
gested itself  that  the  existence  of  such  a  circle  should  be 
established  as  a  preliminary  to  any  investigation  of  parallels. 

An  attempt  in  this  direction  was  made  by  F.  L.  Wachter. 

Wachter,  a  student  under  Gauss  in  Gottingen  [1809], 
and  Professor  of  Mathematics  in  the  Gymnasium  of  Dantzig, 
had  made  several  attempts  at  the  demonstration  of  the  Postu- 
late. He  believed  that  he  had  been  successful,  first  in  a  letter 
to  Gauss  [Dec,  1816],  and  later,  in  a  tract,  printed  at  Dantzig 
in  1817.' 

In  this  pamphlet  he  seeks  to  establish  that  given  any  four 
points  in  space,  (not  on  a  plane),  a  sphere  will  pass  through 
them.    He  makes  use  of  the  following  postulate  : 

Any  four  points  of  space  fully  determifie  a  surface  [the 
surface  of  four  poifits],  and  two  of  these  surfaces  intersect  in  a 
single  line^  completely  determi?icd  by  three  points. 

There  is  no  advantage  in  following  the  argument  by 
means  of  which  Wachter  seeks  to  prove  that  the  surface  of 
four  points  is  a  sphere,  since  he  fails  to  give  a  precise  defini- 
tion of  that  surface  in  his  tract.  His  deductions  have  thus 
only  an  intuitive  character. 

On  the  other  hand  a  passage  in  his  letter  of  1816  de- 
serves special  notice.  It  was  written  after  a  conversation  with 
Gauss,  when  they  had  spoken  of  an  Anti- Euclidean  Geometry. 
In  this  letter  he  speaks  of  the  surface  to  which  a  sphere  tends 


I  Demonstratio  axiomatis  geometrici  in  Euclideis  undechni. 


Wachter  and  Thibaut.  63 

as  its  radius  approaches  infinity,  a  siirface  on  the  Euclidean 
hypothesis  identical  with  a  plane.  He  affirms  that  eveti  in  the 
case  of  the  Fifth  Postulate  being  false,  there  would  be  a  geo- 
metry on  this  surface  identical  with  that  of  the  ordifiary  plane. 
This  statement  is  of  the  greatest  importance  as  it  con- 
tains one  of  the  most  remarkable  results  which  hold  in  the 
system  of  geometry^  corresponding  to  Saccheri's  Hypo- 
thesis of  the  Acute  Angle  [cf.  Lobatschewsky,  §  40].' 

Bernhard  Friedrich  Thibaut  [1775 — 1832]. 

§  30  (bis).  One  other  erroneous  proof  of  the  theorem  that  the 
sum  of  the  angles  of  a  triangle  is  equal  to  two  right  angles  should 
be  mentioned,  since  it  has  recently  been  revived  in  English  textbooks, 
and  to  some  extent  received  official  sanction.  It  depends  upon 
the  idea  of  diredion,  and  assumes  that  translation  and  rotation  are 
independent  operations.  It  is  due  to  Thibaut  [Gì-icndrij]  der  reincn 
Mathetnatik,  2.  Aufl.,  Gottingen,  1809).  Gauss  refers  to  this  "proof" 
in  his  correspondence  with  Schumacher,  and  shows  that  it  involves 
a  proposition  which  not  only  needs  proof,  but  is,  in  essence,  the 
very  proposition  to  be  proved.     Thibaut  argued  as  follows  :  2 — 

"Let  ABC  be  any  triangle  whose  sides  are  traversed  in  order 
from  A  along  AB,  BC,  CA.  While  going  from  ^  to  i?  we  always 
gaze  in  the  direction  ABb  [AB  being  produced  to  b),  but  do  not 
turn  round.  On  arriving  at  B  we  turn  from  the  direction  Bb  by  a 
rotation  through  the  angle  bBC,  until  we  gaze  in  the  direction  BCc. 
Then  we  proceed  in  the  direction  BCc  as  far  as  C,  where  again 
we  turn  from  Cc  to  CAa  through  the  angle  cCA;  and  at  last  arriving 
at  A,  we  turn  from  the  direction  Aa  to  the  first  direction  AB 
through  the  external  angle  aAB,  This  done,  we  have  made  a 
complete  revolution, —  just  as  if,  standing  at  some  point,  we  had 
turned  completely  round;  and  the  measure  of  this  rotation  is  2  ir. 
Hence  the  external  angles  of  the  triangle  add  up  to  2  ir,  and  the 
internal  angles  A-\- B -\- C  =  -n.     Q.  E.  D." 


1  With  regard  to  Wachter,  cf.  P.  StAckel:  Friedrich  Ludwig 
Wachter,  ein  Beitrag  zur  Geschichte  der  nichtetiklidischen  Geometrie. 
Math.  Ann.  Bd.  LIV.  p.  49—85.  (1901).  In  this  article  are  reprinted 
Wachter's  letters  upon  the  subject  and  the  tract  of  1S17  referred 
to  above. 

2  [For  further  discussion  of  this  "proof"  see  W.  B.  Frank- 
LÀNd's  Theories  of  Parallelism,  (Camb. Univ.  Press,  19  lo),  from  which 
this  version  is  taken,  and  Heath's  Euclid,  Vol.  I.,  p.  321.] 


Chapter  III. 
The  Founders  of  Non-Euclidean  Geometry. 

Carl  Friederich  Gauss   [1777 — 1855]. 

§  31.  Twenty  centuries  of  useless  effort,  and  in  particular 
the  last  unsuccessful  investigations  on  the  Fifth  Postulate,  con- 
vinced many  of  the  geometers,  who  flourished  about  the  be- 
ginning of  last  century,  that  the  final  settlement  of  the  theory 
of  parallels  involved  a  problem  whose  solution  was  impossible. 
The  Gottingen  school  had  officially  declared  the  necessity 
of  admitting  the  Euclidean  hypothesis.  This  view,  expressed 
by  Klugel  in  his  Conatuum  [cf  p.  44]  was  accepted  and  sup- 
ported by  his  teacher,  A.  G.  Kastner,  then  Professor  in  the 
University  of  Gottingen.^ 

Nevertheless  keen  interest  was  always  taken  in  the 
subject;  an  interest  which  still  continued  to  provide  those 
who  sought  for  a  proof  of  the  postulate  with  fruitless  labour, 
and  led  finally  to  the  discovery  of  new  systems  of  geometry. 
These,  founded  like  ordinary  geometry  on  intuition,  extend 
into  a  far  wider  field,  freed  from  the  principle  embodied  in 
the  Euclidean  Postulate. 

How  difficult  was  this  advance  towards  the  new  order 
of  ideas  will  be  clear  to  any  one  who  carries  himself  back  to 
that  period,  and  remembers  the  trend  of  the  Kantian  Philo- 
sophy, then  predominant. 

§  32.  Gauss  was  the  first  to  have  a  clear  view  of  a 
geometry  independent  of  the  Fifth  Postulate,   but  this   re- 


I  Cf.  Enc.el  u.  StAckel:  Tit.  der  P.  p.  139—142, 


Gauss  and  W.  Bolyai.  gc 

mained  for  quite  fifty'  years  concealed  in  the  mind  of  the 
great  geometer,  and  was  only  revealed  after  the  works  of 
LoBATSCHEWSKY  [1829 — 30]  and  J.  Bolyai  [1832]  appeared. 

The  documents  which  allow  an  approximate  reconstruct- 
ion of  the  lines  of  research  followed  by  Gauss  in  his  work 
on  parallels,  are  his  correspondence  with  W.  Bolyai,  Olbers, 
Schumacher,  Gerling,  Taurinus  and  Bessel  [1799 — 1844]; 
two  short  articles  in  the  Goti,  gelehrten  Anzeigm{\2>i6^  1822]; 
and  some  notes  found  among  his  papers,  [1831].^ 

Comparing  the  various  passages  in  Gauss's  letters,  we 
can  fix  the  year  1792  as  the  date  at  which  he  began  his  'Med- 
itations'. 

The  following  portion  of  a  letter  to  W.  Bolyai  [Dec.  1 7, 
1799]  proves  that  Gauss,  Hke  Saccheri  and  Lambert  before 
him,  had  attempted  to  prove  the  truth  of  Postulate  V.  by  as- 
suming it  to  be  false. 

'As  for  me,  I  have  already  made  some  progress  in  my 
work.  However  the  path  I  have  chosen  does  not  lead  at 
all  to  the  goal  which  we  seek,  and  which  you  assure  me  you 
have  reached.3  It  seems  rather  to  compel  me  to  doubt  the 
truth  of  geometry  itself. 

'It  is  true  that  I  have  come  upon  much  which  by  most 
people  would  be  held  to  constitute  a  proof:  but  in  my  eyes 
it  proves  as  good  as  nothing.  For  example,  if  one  could 
show  that  a  rectilinear  triangle  is  possible,  whose  area  would 
be  greater  than  any  given  area,  then  I  would  be  ready  to 
prove  the  whole  of  geometry  absolutely  rigorously. 

'Most  people  would  certainly  let  this  stand  as  an  Axiom; 
but  I,  no!    It  would,  indeed,  be  possible  that  the  area  might 


1  [It  would  be  more  correct  to  say  over  thirty.] 

2  Cf.  Gauss,  Werke,  Bd.  VIE.  p.  157—268. 

3  It  is  to  be  remembered  that  W.  Bolyai  was  working  at 
this  subject  in  Gottingen  and  thought  he  had  overcome  his  diffi- 
culties.    Cf.  3  29. 

5 


^S  III.     The  Founders  of  Non-Euclidean  Geometry. 

always  remain  below  a  certain  limit,  however  far  apart  the 
three  angular  points  of  the  triangle  were  taken.' 

In  1804,  replying  to  W.  Bolyai  on  his  Theoria  parall- 
elamm,  he  expresses  the  hope  that  the  obstacles  by  which 
their  investigations  had  been  brought  to  a  standstill  would 
finally  leave  a  way  of  advance  open.^ 

From  all  this,  Stackel  and  Engel,  who  collected  and 
verified  Gauss's  correspondence  on  this  subject,  come  to  the 
conclusion  that  the  great  geometer  did  not  recognize  the 
existence  of  a  logically  sound  Non-Euclidean  geometry  by 
intuition  or  by  a  flash  of  genius  :  that,  on  the  contrary,  he 
had  spent  upon  this  subject  many  laborious  hours  before  he 
had  overcome  the  inherited  prejudice  against  it. 

Did  Gauss,  when  he  began  his  investigations,  know  the 
writings  of  Saccheri  and  Lambert?  What  influence  did  they 
exert  upon  his  work?  Segre,  in  his  Congetture^  already  re- 
ferred to  [p.  44  note  2],  remarks  that  both  Gauss  and  W. 
Bolyai,  while  students  at  Gottingen,  the  former  from  1795 
— 98,  the  later  from  1796 — 99,  were  interested  in  the  theory 
of  parallels.  It  is  therefore  possible  that,  through  Kastner 
and  Seyffer,  who  were  both  deeply  versed  in  this  subject 
they  had  obtained  knowledge  both  of  the  Euclides  ab  omni 
naevo  vindicatus  and  of  the  Theorie  der  Faralleiiinien.  But 
the  dates  of  which  we  are  certain,  although  they  do  not  con- 
tradict this  view,  fail  to  confirm  it  absolutely. 

§  33.  To  this  first  period  of  Gauss's  work,  after  1 8 1 3 
there  follows  a  second.  Of  it  we  obtain  some  knowledge 
chiefly  from  a  few  letters,  one  written  by  Wachter  to  Gauss 
[18 1 6];  others  [sent  |by  Gauss  to  Gerling  [i8i9],jTaurinus 
[1824]  and  Schumacher  [183 i];  and  also  from  some  notes 
found  among  Gauss's  papers. 


I  [It    should    be    noticed  that  these  efforts  were  still  directed 
towards  proving  the  truth  of  Euclid's  postulate.] 


Gauss's  "Meditations". 


^ 


These  documents  show  us  that  Gauss,  in  this  second 
period,  had  overcome  his  doubts,  and  proceeded  with  his  de- 
velopment of  the  fundamental  theorems  of  a  new  geometry, 
which  he  first  czWs,  Anti-Euclidean  [cf.WACHTER's  letter  quoted 
on  p.  62];  then  Astral  Geometry  [following  Schweikart,  cf. 
p.  76];  dina  ^nsWy,  Non-Euclidean  [cf  letter  to  Schumacher]. 
Thus  he  became  convinced  that  the  Non-Euclidean  Geometry 
did  not  in  itself  involve  any  contradiction,  though  at  first 
sight  some  of  its  results  had  the  appearance  of  paradoxes 
[letter  to  Schumacher,  July  12,  183 1]. 

However  Gauss  did  not  let  any  rumour  of  his  opinions 
get  abroad,  being  certain  that  he  would  be  misunderstood. 
[He  was  afraid  of  the  clamour  of  the  Boeotiatis;  letter  to  Bessel, 
Jan.  27,  1829].  Only  to  a  few  trusted  friends  did  he  reveal 
something  of  his  work.  When  circumstances  compel  him  to 
write  to  Taurinus  [1824]  on  the  subject,  he  begs  him  to 
keep  silence  as  to  the  information  which  he  imparted  to  him. 

The  notes  found  among  Gauss's  papers  contain  two 
brief  synopses  of  the  new  theory  of  parallels,  and  probably 
belong  to  the  projected  exposition  of  the  Non-Euclidean  Geo 
metry,  with  regard  to  which  he  wrote  to  Schumacher  [on 
May  17,  1 831]:  *In  the  last  few  weeks  I  have  begun  to  put 
down  a  few  of  my  own  Meditations,  which  are  already  to 
some  extent  nearly  40  years  old.  These  I  had  never  put  in 
writing,  so  that  I  have  been  compelled  three  or  four  times 
to  go  over  the  whole  matter  afresh  in  my  head.  Also  I  wished 
that  it  should  not  perish  with  me.' 

§  34.    Gauss  defines  parallels  as  follows  :  ^ 
If  the  coplanar  straight  lines  AM,  BN,  do  not  intersect 
each  ether,  while,  on  the  other  hand,  every  straight  line  through 

I  [In  this  section  upon  Gauss's  work  on  Parallels  fuller  use 
has  been  made  of  the  material  in  his  Collected  Works  (Gauss, 
Werke,  Bd.  VIII,  p.  202—9)]. 

S* 


Fig.  33- 


68  III.     The  Founders  of  Non-Euclidean  Geometry. 

A  between  AM  and  AB  cuts  BN,  then  AM  is  said  to  be  paral- 
lel to  BN{^g.  ii\ 

He   supposes    a    straight 
B  !—-.__  line  passing  through  A^   to 

start  from  the  position  AB, 
and  then  to  rotate  continu- 
ously on  the  side  towards 
^^^  which  BN  is  drawn,  till  it 
reaches  the  position  AC^  in 
Cèjt  BA  produced.    This  line  be- 

gins by  cutting  j^iVand  in  the 
end  it  does  not  cut  it.  Thus 
there  can  be  one  and  only 
one  position,  separating  the  lines  which  intersect  ^iVfrom 
those  which  do  not  intersect  it.  This  must  be  "ùxt  first  of  the 
lines,  which  do  not  cut  BN:  and  thus  from  our  definition  it 
is  the  parallel  AM)  since  there  can  obviously  be  no  last  line 
of  the  set  of  lines  which  intersect  BN. 

It  will  be  seen  in  what  way  this  definition  differs  from 
Euclid's.  If  Euclid's  Postulate  is  rejected,  there  could  be  dif- 
ferent lines  through  A,  on  the  side  towards  which  BN  is 
drawn,  which  would  not  cut  BN.  These  lines  would  all  be 
parallels  to  BN  according  to  Euclid's  Definition.  In  Gauss's 
definition  only  the  first  of  these  is  said  to  be  parallel 
\.oBN. 

Proceeding  with  his  argument  Gauss  now  points  out 
that  in  his  definition  the  starting  points  of  the  lines  AM  and 
BN  are  assumed,  though  the  lines  are  supposed  to  be  pro- 
duced indefinitely  in  the  directions  of  AM  and  BN. 

I.  He  proceeds  to  show  that  the  parallelism  of  the  line 
AM  to  the  line  BN  is  independent  of  the  points  A  a?id  B,  pro- 
vided the  sense  in  which  the  lines  are  to  be  produced  indefinitely 
remain  the  same. 

It  is  obvious  that  we  would  obtain  the  same  parallel  AM 


Gauss's  Theory  of  Parallels. 


69 


if  we  kept  A  fixed  and  took  instead  of  B  another  point  B' 
on  the  line  BN,  or  on  that  Hne  produced  backwards. 

It  remains  to  prove  that  \i  AMis  parallel  to  BJV (or  the 
point  A,  it  is  also  the  parallel  to  BNiox  any  point  upon  AM, 
or  upon  AM  produced  backwards. 

Instead  of  ^  [Fig.  34]  take  another  starting  point  A'  upon 
AM.     Through  A\  between 
A'B  and  A'M,  draw  the  line 
A'F  in  any  direction.  B|< 

Through  Q,  any  point  on 
A'F,  between  A'  and  F,  draw 
the  line  AQ. 

Then,  from  the  definition,  A  ■ 
AQ  must  cut  BN,  so  that  it 
is  clear    QF  must  also   cut 
BN. 

Thus  AA'M  is  the  first  of 
the  lines  which  do  not  cut  BN,  and  A'M  is  parallel  to  BN. 
Again  take  the  point  A'  upon  AM  produced  backwards 
[Fig-  35]- 


^M 


Fig-  34. 


Fig.  35- 


Draw  through  A',  between  A'B  and  A'M,  the  line  A'F 
in  any  direction. 

Produce  A'F  backwards  and  upon  it  take  any  point  Q. 
Then,  by  the  definition,  QA  must  cut  BN,  for  example, 


70 


III.     The  Founders  of  Non-Euclidean  Geometry. 


in  R.    Therefore  AP  lies  within  the  closed  figure  AARB, 
and  must  cut  one  of  the  four  sides  AA^  AR,  RB,  and  BA. 
Obviously  this  must  be  the  third  side  RB,  and  therefore 
AM  is  parallel  to  BN. 

II.  The  Reciprocity  of  the  Parallelism  can  also  be  estab- 
lished. 

In  other  words,  if  AM  is  parallel  to  BN,  then  BN  is 
also  parallel  to  AM. 

Gauss  proves  this  result  as  follows: 

From  any  point  B  upon  BN  draw  BA  perpendicular  to 
AM.   Through  B  draw  any  line  BN'  between  BA  and  BN. 

At  B,  on  the  same  side  of  AB  as  BN,  make 
<^  ABC^  V2  ^N'BN 
There  are  two  possible  cases: 

Case  (i),  when  BC  cuts  AM  [cf.  Fig.  36]. 

Case  (ii),  when  BC  does  not  cut  AM  [cf.  Fig.  37]. 


Fig.  36. 

Case  (i).  Let  BC  cut  AM  in  D.  Take  AE  =  AD,  and 
join  BE.    Make  ^BDF^^BED. 

Since  AM  is  parallel  to  BN^  DF  must  cut  BM^  for 
example,  in  G. 

From  EM  cut  off  EH  equal  to  DG. 

Then,  in  the  triangles  BEH  s^nà  BDG,  it  follows  that 


Gauss's  Theory  of  Parallels  (contd.). 


71 


JM. 


Therefore  «^  EBD  =  ^HBG. 
But  <^  EBD  =  ^N'BN. 
Therefore  BJV  and  BII  coincide,  and  BN'  must  cut 

But  BN"  is  any  line  through  B,  between  BA  and  BN. 
Therefore  BN  is  parallel  to  AM. 

B 


Fig.  37- 

Case  (ii).   In  this  case  let  Z>  be  any  arbitrary  point  upon 
AM.   Then  with  the  same  argument  as  above, 
■^  EBB  =  <^  GBH, 

But  ^  ABD  <  <  ^^C. 

Therefore  <^  ^^Z>  <  ^  iV^'^iV. 

Therefore  <^  GBH<.^N'BN. 

Therefore  BN'  must  cut  AM. 

But  ^iV"  is  any  line  through  B,  between  BA  and  BN. 

Therefore  BN  is  parallel  to  AM. 

Thus  in  both  cases  we  have  proved  that  \iAM\5  parallel 
to  BN,  then  BN  is  parallel  to  AM.  "■ 

The  next  theorem  proved  by  Gauss  in  this  synopsis  is 
as  follows: 


[I  Gauss's  second  proof  of  this  theorem  is  given  in  the  German 
translation.  However  it  will  be  found  that  in  it  he  assumes  that  BC 
cuts  AM,  and  to  prove  this  the  argument  used  above  is  necessary.] 


72  III-     The  Founders  of  Non-Euclidean  Geometry. 

III.    If  the  line  (i)  is  parallel  to  the  line  (2)  arid  to  the 
line  (3),  then  (2)  and  (3)  are  parallel  to  each  other. 

Case  (i).    Let  the  line  (i)  lie  between  (2)  and  (3)  [cf. 

Fig.  38]. 

Let  A  and  B  be  two  points  on  (2)  and  (3),  and  let  AB 
cut  (i)  in  C. 

Through  A  let  an  arbitrary  line  AD  be  drawn  between 

AB  and  (2).  Then  it  must 
^é: ^  cut  (i),  and  on  being  pro- 

duced must  also  cut  (3). 

Since  this  holds  for  every 
line  such  as  AD,  (2)  is 
parallel  to  (3). 

Case  (ii).    Let  the  line 

(i)  be  outside  both  (2)  and 

(3),  and  let  (2)  He  between 

(i)  and  (3)  [cf.  Fig.  39]. 

If  (2)  is  not  parallel  to  (3),  through  any  point  chosen  at 

random  upon  (3),  a  line  different  from  (3)  can  be  drawn 

which  is  parallel  to  (2). 

This,  by  Case  (i),  is  also  par- 
allel to  (i),  which  is  absurd. 
This  short  Note  on  Parall- 
els closes  with  the  theorem 
that  if  tivo  lities  AM  and  BN 
are  parallel,  these  lines  produced 
backwards  cannot  tneet. 
From  all  this  it  is  evident  that  the  parallelism  of  Gauss 
xtitzxis  parallelism  in  a  given  sense.  Indeed  his  definition  of 
parallels  deals  with  a  line  drawn  from  A  on  a.  definite  side  of 
the  transversal  AB:  e.  g.,  the  ray  drawn  to  the  right,  so  that 
we  might  speak  of  AM  as  the  parallel  to  BJV  towards  the  right. 
The  parallel  from  A  to  BJV  towards  the  left  is  not  necessari- 
ly AM.   If  it  were,  we  would  obtain  the  Euclidean  hypothesis. 


Fig.  38. 


?•  39- 


Corresponding  Points. 


n 


The  two  lines,  in  the  third  theorem,  which  are  each  pa- 
rallel to  a  third  line,  are  thus  both  parallels  in  the  same  sense 
(both  left-hand,  or  both  right-hand  parallels). 

In  a  second  memorandum  on  parallels,  Gauss  goes  over 
the  same  ground,  but  adds  the  idea  of  Corresponding  Points 
on  two  parallels  AA ,  BB' .  Two  points  A,  B  are  said  to  corre- 
spond^ when  AB  makes  equal  internal  angles  with  the  parallels 
en  the  same  side  [cf.  Fig.  40]. 


Fig.  40. 


Fig.  41. 


With  regard  to  these  Corresponding  Points  he  states  the 
following  theorems: 

(i)  If  A,  B  are  two  correspofiding  points  upon  tivo  paral- 
lels, and  M  is  the  middle  poitit  of  AB,  the  line  MN,  perpen- 
dicular to  AB,  is  parallel  to  the  two  given  lines,  and  every 
point  on  the  same  side  of  MN  as  A  is  nearer  A  than  B. 

(ii)  If  A,  B  are  two  corresponding  points  upon  the 
parallels  {\)  and  {2),  and  A',  B'  two  other  correspo7iding  points 
on  the  same  lifies,  then  AA  =  BB',  and  co?iversely. 

(iii)  If  A,  B,  C  are  three  points  on  the  parallels  (i),  (2) 
and  (3),  such  that  A  and  B,  B  and  C,  correspond,  then  A  and 
C  also  correspond. 


>jA  III.     The  Founders  of  Non-Euclidean  Geometry. 

The  idea  of  Corresponding  Points,  when  taken  in  con- 
nection with  three  Hnes  of  a  pencil  (that  is,  three  concurrent 
lines  [cf.  Fig.  41]  allows  us  to  define  the  circle  as  the  locus  of 
the  points  on  the  lines  of  a  pencil  which  correspond  to  a  given 
point.  But  this  locus  can  also  be  constructed  when  the  lines 
of  the  pencil  are  parallel.  In  the  Euclidean  case  the  locus 
is  a  straight  line  :  but  putting  aside  the  Euclidean  hypothesis, 
the  locus  in  question  is  a  line,  having  many  properties  in 
common  with  the  circle,  but  yet  not  itself  a  circle.  Indeed  if 
any  three  points  are  taken  upon  it,  a  circle  cannot  be  drawn 
through  them.  This  line  can  be  regarded  as  the  limiting  case 
of  a  circle,  when  its  radius  becomes  infinite.  In  the  Non- 
Euclidean  geometry  of  Lobatschewsky  and  Bolyai,  this  locus 
plays  a  most  important  part,  and  we  shall  meet  it  there  under 
the  name  of  the  Horocycle.' 

This  work  Gauss  did  not  need  to  complete,  for  in  1832 
he  received  from  Wolfgang  Bolyai  a  copy  of  the  work  of 
his  son  Johann  on  Absolute  Geometry. 

From  letters  before  and  after  the  date  at  which  he 
interrupted  his  work,  we  know  that  Gauss  had  discovered  in 
his  geometry  an  Absolute  Unit  of  Length  [cf.  Lambert  and 
Legendre],  and  that  a  constant  k  appeared  in  his  formulae, 
by  means  of  which  all  the  problems  of  the  Non-Euclidean 
Geometry  could  be  solved  [letter  to  Taurinus,  Nov.  8, 
1824]. 

Speaking  more  fully  of  these  matters  in  1831  [letter  to 


I  [Lobatschewsky  ;  Gremkreis,  Courbe-Umite  or  Iloricycle.  BOL- 
YAI;  Parazykl,  L-lÌ7iie. 

It  is  interesting  to  notice  that  Gauss,  even  at  this  date, 
seems  to  have  anticipated  the  importance  of  the  Ilorocycle.  The 
definition  of  Corresponding  Points  and  the  statement  of  their 
properties  is  evidently  meant  to  form  an  introduction  to  the  dis- 
cussion of  the  properties  of  this  curve,  to  which  he  seems  to  have 
given  the  name  Trope.'] 


The  Perimeter  of  a  Circle.  7C 

Schumacher],  he  gave  the  length  of  the  circumference  of  a 
circle  of  radius  r  in  the  form 


■nk\e^—e    ^) . 


With  regard  to  k,  he  says  that,  if  we  wish  to  make  the  new 
geometry  agree  wth  the  facts  of  experience,  we  must  suppose 
k  infinitely  great  in  comparison  with  all  known  measurements. 
For  >è  ^  00 ,  Gauss's  expression  takes  the  usual  form 
for  the  perimeter  of  a  circle.  '  The  same  remark  holds  for  the 
whole  of  Gauss's  system  of  geometry.  It  contains  Euclid's 
system,  as  the  limiting  case,  when  /è  =  00  .  ^ 

Ferdinand  Karl  Schweikart  [1780 — 1859]. 

§  35.  The  investigations  of  the  Professor  of  Jurispru- 
dence, F.  K.  ScHWEiKART,3  date  from  the  same  period  as 
those  of  Gauss,  but  are  independent  of  them.  In  1807  he 
published  Die  Theorie  der  Parallellinien  nebst  dem  Vorschlage 
ihrer  Verbannung  aus  der  Geometrie.  Contrary  to  what  one 
might  expect  from  its  title,  this  work  does  not  contain  a 
treatment  of  parallels  independent  of  the  Fifth  Postulate, 
but  one  based  on  the  idea  of  the  parallelogram. 

But  at  a  later  date,  Schweikart,  having  discovered  a 
new  order  of  ideas,  developed  a  geometry  independent  of 
Euclid s  hypothesis.  When  in  Marburg  in  December,  1818, 
he  handed  the  following  memorandum  to  his  colleague  Ger- 
LiNG,  asking  him  to  communicate  it  to  Gauss  and  obtain  his 
opinion  upon  it: 


1  To   show  this  we  need  only  use  the  exponential  series. 

2  For  other  investigations  by  Gauss,  cf.  Note  on  p.  90. 

3  He  studied  law  at  Marburg  and  from  1796 — 98  attended  the 
lectures  on  Mathematics  given  in  that  University  by  Professor  J.  K, 
F.  Hauff,  the  author  of  various  memoirs  on  parallels,  cf.  Th.  der 
P.  p.  243. 


n^  III.     The  Founders  of  Non-Euclidean  Geometry. 

Memorandum. 

'There  are  two  kinds  of  geometry— a  geometry  in  the 
strict  sense — the  Eudidean;  and  an  astral  geometry  [astra- 
hsche  Grofienlehre]. 

'Triangles  in  the  latter  have  the  property  that  the  sum 
of  their  three  angles  is  not  equal  to  two  right  angles.' 

'This  being  assumed,  we  can  prove  rigorously: 

a)  That  the  sum  of  the  three  angles  of  a  triangle  is  less 
than  two  right  angles; 

b)  that  the  sum  becomes  ever  less,  the  greater  the  area 
of  the  triangle; 

c)  that  the  altitude  of  an  isosceles  right-angled  triangle 
continually  grows,  as  the  sides  increase,  but  it  can 
never  become  greater  than  a  certain  length,  which 
I  call  the  Cofistant. 

Squares  have,  therefore,  the  following  form  [Fig.  42]. 
'If  this  Constant  were  for  us  the  Radius  of  the  Earth, 
(so  that  every  line  drawn  in  the 
universe  from  one  fixed  Star 
to  another,  distant  90°  from  the 
first,  would  be  a  tangent  to  the 
surface  of  the  earth),  it  would  be 
infinitely  great  in  comparison  with 
the  spaces  which  occur  in  daily 
life. 

'The  Euclidean  geometry  holds 
only  on  the  assumption  that  the 
Constant  is  infinite.  Only  in  this 
case  is  it  true  that  the  three  angles  of  every  triangle  are  equal 
to  two  right  angles:  and  this  can  easily  be  proved,  as  soon 
as  we  admit  that  the  Constant  is  infinite.'  ^ 

Schweikart's  Astral  Geometry  and  Gauss's  Non-Euclid- 


Fig.  42. 


Schweikart's  Work.  nj 

ean  Geometry  exactly  correspond  to  the  systems  of  Sac- 
CHERi  and  Lambert  for  the  Hypothesis  of  the  Acute  Angle. 
Indeed  the  contents  of  the  above  memorandum  can  be  ob- 
tained directly  from  the  theorems  of  Saccheri,  stated  in 
Klùgel's  Conatuum,  and  from  Lambert's  Theorem  on  the 
area  of  a  triangle.  Also  since  Schweikart  in  his  Theorie  of 
1807  mentions  the  works  of  the  two  latter  authors,  the  direct 
influence  of  Lambert,  and,  at  least,  the  indirect  influence  of 
Saccheri  upon  his  investigations  are  established.^ 

In  March,  1 8 1 9  Gauss  replied  to  Gerling  with  regard 
to  the  Astral  Geometry.  He  compliments  Schweikart,  and 
declares  his  agreement  with  all  that  the  sheet  of  paper  sent 
to  him  contained.  He  adds  that  he  had  extended  the  Astral 
Geometry  so  far  that  he  could  completely  solve  all  its  pro- 
blems, if  only  Schweikart's  Constant  were  given.  In  con- 
clusion, he  gives  the  upper  limit  for  the  area  of  a  triangle 
in  the  form  J 

[log  hyp  (I  +  \2)Y  ' 
Schweikart  did  not  publish  his  investigations. 

Franz  Adolf  Taurinus  [1794 — 1874]. 
§  36.    In  addition  to  carrying  on  his  own  investigations 
on  parallels,  Schweikart  had  persuaded  [1820]  his  nephew 
Taurinus  to  devote  himself  to  the  subject,  calling  his  atten- 


1  Cf.  Gauss,  Werke,  Bd.  VIII,  p.  iSo— 181. 

2  Cf.  Segre's  Congetture,  cited  above  on  p.  44. 

3  The  constant  which  appears  in  this  formula  is  Schweikart's 
Constant  C,  not  Gauss-'s  constant  /',  in  terms  of  which  he  expressed 
the  length  of  the  circumference  of  a  circle,  (cf.  p.  75).  The  two 
constants  are  connected  by  the  following  equation: 

log    (1+1/2)- 


78  in.     The  Founders  of  Non-Euclidean  Geometry. 

tion  to  the  Astral  Geometry,  and  to  Gauss's  favourable  ver- 
dict upon  it. 

Taurinus  appears  to  have  taken  up  the  subject  seriously 
for  the  first  time  in  1824,  but  with  views  very  different  from 
his  uncle's.  He  was  then  convinced  of  the  absolute  truth  of 
the  Fifth  Postulate,  and  always  remained  so,  and  he  cherish- 
ed the  hope  of  being  able  to  prove  it.  FaiHng  in  his  first  at- 
tempts, under  the  influence  of  Gauss  and  Schweikart,  he 
again  began  the  study  of  the  question.  In  1825  he  publish- 
ed a  Theorie  der  Parallellinien^  containing  a  treatment  of  the 
subject  on  Non-Euclidean  lines,  the  rejection  oi  the  Hypothesis 
of  the  Obtuse  Angle,  and  some  investigations  resembling  those 
of  Saccheri  and  Lambert  on  the  Hypothesis  of  the  Acute 
Angle.  He  found  in  this  way  Schweikart's  Constant,  which 
he  called  a  Parameter.  He  thought  an  absolute  unit  of 
length  impossible,  and  concluded  that  all  the  systems,  corre- 
sponding to  the  infinite  number  of  values  of  the  parameter, 
ought  to  hold  simultaneously.  But  this,  in  its  turn,  led  to  con- 
siderations incompatible  with  his  conception  of  space,  and 
thus  Taurinus  was  led  to  reject  the  Hypothesis  of  the  Acute 
Angle  while  recognising  the  logical  compatibility  of  the  propo- 
sitions which  followed  from  it. 

In  the  next  year  Taurinus  published  his  Geometriae  Pri- 
ma Elementa  [Cologne,  1826],  in  which  he  gave  an  improved 
version  of  his  researches  of  1825.  This  work  concludes  with 
a  most  important  appendix,  in  which  the  author  shows  how 
a  system  of  analytical  geometry  could  be  actually  constructed 
on  the  Hypothesis  of  the  Acute  Angle.  ^ 

With  this  aim  Tauriuus  starts  from  the  fundamental  for- 
mula of  Spherical  Trigonometry — 


I  For  the  final  influence  of  Saccheri  and  Lambert  upon  Tau- 
rinus, cf.  SeGRE's  Congetture,  quoted  above  on  p.  44. 


The  Work  of  Taurinus.  yg 

a  b  C.Ò.C  . 

COS  -r  =  COS  -7  COS  -r  +  sm  ^  sm  -,    cos  A, 

In  it  he  transforms  the  real  radius  k  into  the  imaginary  radius 

ik.    Using  the  notation  of  the  hyperboHc  functions,  we  thus 

have 

(i)        cosh  -T   =  cosh  —  cosh  — sinh  —  sinh  -^  cos  A. 

This  is  the  fundamental  formula  of  the  Logarithmic- 
Spherical  Geometry  \logarithmisch-spharischen  Geometrie'\  of 
Taurinus. 

It  is  easy  to  show  that  in  this  geometry  the  sum  of  the 
angles  of  a  triangle  is  less  than  180°.  For  simplicity  we  take 
the  case  of  an  equilateral  triangle,  putting  a=b=c  in  (i). 

Solving,  for  cos  A,  we  obtain 

cosh  — 
(i*)  cos  ^  = 


cosh—  +  I 


But  sech  T<C  I- 


Therefore  cos  ^  ]>  ^/a- 

Thus  A  is  less  than  60°,  and  the  sum  of  the  angles  of 
the  triangle  is  less  than  180°. 

It  is  instructive  to  note,  that,  from  (i*). 
Lt.  (cos  A)  =  Vz. 

a  ==  o 

So  that  in  the  Hmit  when  a  becomes  zero,  A  is  equal  to  60°. 
Therefore,  in  the  log. -spherical geotnetry,  the  sum  of  the  angles 
of  a  triangle  tends  to  x8o°  when  the  sides  tend  to  zero. 
We  may  also  note  that  from  (i*) 

Lt.  (cos  A)  =  V2  ; 

k       « 
so  that  in  the  limit  when  k  is  infinite,  A  is  equal  to  60°.  There- 
fore, when  the  constant  k  tends  to  infinity,  the  angles  of  the 
equilateral  triangle  are  each  equal  to  60°,  as  in  the  ordinary 
geometry. 


8o  ni.     The  Founders  of  Non-Euclidean  Geometry. 

More  generally,  using  the  exponential  forms  for  the  hy- 
perbolic functions,  it  will  be  seen  that  in  the  limit  when  k  is 
infinite  (i)  becomes 

a^  =  b"^  -^  c-  —  2bc  cos  A, 

the  fundamental  formula  of  Euclidean  Plane  Trigonometry. 

§  37.  The  second  fundamental  formula  of  Spherical 
Trigonometry, 

cos  A  =  —  cos  B  cos  C  +  sin  ^  sin  C  cos  -y  > 

by  simply  interchanging  the  cosine  with  the  hyperbolic  cosine, 

gives  rise  to  the  second  fundamental  formula  of  the  log.-spher- 

zVa/ geometry: 

a, 

(2)  COS  A  =  —  cos  B  cos  C  +  sin  B  sin  C  cosh  -r. 

For  A  =  o  and  C=  90°,  we  have 

(3)  cosh  X  =   •'  ^" 
^■^  k         sin  B 

The  triangle  corresponding  to  this  formula  has  one  angle 
zero  and  the  two  sides  containing  it  are  infinite  and  parallel 
[asymptotic].  [Fig.  43.]  The  angle  B^  between  the  side  which 


Fig.  43- 

is  parallel  and  the  side  which  is  perpendicular  to  CA,  is  seen 
from  (3)  to  be  a  function  of  a.  From  this  onward  we  can 
call  it  the  Angle  of  Parallelism  for  the  distance  a  [cf.  Lobat- 
SCHEWSKY,  p.  87]. 

For  B  =  45°,  the  segment  BC^  which  is  given  by  (3),  is 
Schweikart's  Constant  [cf.  p.  76].    Thus,  denoting  it  by  P, 


The  Angle  of  Parallelism.  gl 

cosh  ^  =  V2, 
from  which,  solving  for  k,  we  have 

k^ ^-_. 

log  (I  +  V2) 

This  relation  connecting  the  two  constants  /'  and  ^  was 
given  by  Taurinus.  The  constant  k  is  the  same  as  that  em- 
ployed by  Gauss  [cf.  p.  75]  in  finding  the  length  of  the  cir- 
cumference of  a  circle. 

§  38.  Taurinus  deduced  other  important  theorems  in 
the  log.-spherical  geometry  by  further  transformations  of  the 
formulae  of  Spherical  Trigonometry,  replacing  the  real  radius 
by  an  imaginary  one. 

For  example,  that  the  area  of  a  triangle  is  proportional 
to  its  defect  [Lambert,  p.  46]  : 

that  the  superior  limit  of  that  area  is 

„     ,   ,  -.,-.,,  [Gauss,  p.  77  ; 

[log(l-{-^2)]2 

that  the  length  of  the  circumference  of  a  circle  of  radius  r  is 

2Tr/è  sinh  -.  [Gauss,  p.  75]; 
that  the  area  of  a  circle  of  radius  r  is 

2TtZ'^  (cosh  -T-  —  i); 

that  the  area  of  the  surface  of  a  sphere  and  its  volume,  are 
respectively 

■y 

47T/&^  smh^  -,, 

and  2TT/è3  (sinh  ,   cosh  y  —  — ). 

We  shall  not  devote  more  space  to  the  different  anaiyt- 

6 


32  III-     The  Founders  of  Non-Euclidean  Geometry. 

ical  developments,  since  a  fuller  discussion  would  cast  no 
fresh  light  upon  the  method.  However  we  note  that  the 
results  of  Taurinus  confirm  the  prophecy  of  Lambert  on 
the  Third  Hypothesis  [cf.  p.  50],  since  the  formulae  of  the 
log.-spherical  geometry,  interpreted  analytically,  give  the  fun- 
damental relations  between  the  elements  of  a  triangle  traced 
upon  a  sphere  of  imaginary  radius.^ 

To  this  we  add  that  Taurinus  in  common  with  Lambert 
recognized  that  Spherical  Geometry  corresponds  exactly  to 
the  system  valid  in  the  case  of  the  Hypothesis  of  the  Obtuse 
Angle:  further  that  the  ordinary  geometry  forms  a  hnk  be- 
tween spherical  geometry  and  the  log.-spherical geometry. 

Indeed,  if  the  radius  k  passes  continuously  from  the  real 
domain  to  the  purely  imaginary  one,  through  infinity,  we  pro- 
ceed from  the  spherical  system  to  the  log.- spherical  system, 
through  the  Euclidean. 

Although  Taurinus,  as  we  have  already  remarked,  ex- 
cluded the  possibility  that  a  log.-spherical geometry  could  be 
vahd  on  the  plane,  the  theoretical  interest,  which  it  offers, 
did  not  escape  his  notice.  Calling  the  attention  of  geo- 
meters to  his  formulae,  he  seemed  to  prophecy  the  existence 


I  At  this  stage  it  should  be  remarked  that  Lambert,  simul- 
taneously with  his  researches  on  parallels,  was  working  at  the  tri- 
gonometrical functions  with  an  imaginary  argument,  whose  connection 
with  Non-Euclidean  Geometry  was  brought  to  light  by  Taukinus. 
Perhaps  Lambert  recognised  that  the  formulae  of  Spherical  Trig- 
onometry were  still  real,  even  when  the  real  radius  was  changed 
in  a  purely  imaginary  one.  In  this  case  his  prophecy  with  regard 
to  the  Hypothesis  of  the  Acute  Angle  (cf.  p.  50)  would  have  a  firm 
foundation  in  his  own  work.  However  we  have  no  authority  for 
the  view  that  he  had  ever  actually  compared  his  investigations  on 
the  trigonometrical  functions  with  those  on  the  theory  of  parallels. 
Cf.  P.  StAckel:  Bcmerkungen  sit  Lamberts  Theorie  der  Parallellinien. 
Biblioteca  Math.  p.  107 — lio.     (1899). 


Some  Conclusions  by  Taurinus.  83 

of  some  concrete  case  in  which  they  would  find  an  inter- 
pretation. * 


I  The  important  service  rendered  by  Schweikart  and  Tau- 
rinus towards  the  discovery  of  the  Non-Euclidean  Geometry  was 
recognised  and  made  known  by  Engel  and  Stackel.  In  their 
Th.  der  P.,  they  devote  a  whole  chapter  to  those  authors,  and 
quote  the  most  important  passages  in  Taurinus'  writings,  besides 
some  letters  which  passed  between  him,  Gauss  and  Schweikart. 
Cf.  Stackel:  Franz  Adolf  Taurinus,  Abhandl.  zur  Geschichte  der 
Math.,  IX,  p.  397 — 427  (1899). 


Chapter  IV. 

The  Founders  of  Non-Euclidean  Geometry 
(Contd.). 

Nicolai  Ivanovitsch  Lobatschewsky  [1793 — 1856],' 

§  39.  Lobatschewsky  studied  mathematics  at  the  Uni- 
versity of  Kasan  under  a  German  J.  M.  C.  Bartels  [1769 — 
1836],  who  was  a  friend  and  fellow  countryman  of  Gauss. 
He  took  his  degree  in  18 13  and  remained  in  the  University, 
first  as  Assistant,  and  then  as  Professor.  In  the  latter  position 
he  lectured  upon  mathematics  in  all  its  branches  and  also 
upon  physics  and  astronomy. 

As  early  as  181 5  Lobatschewsky  was  working  at  paral- 
lels, and  in  a  copy  of  his  notes  for  his  lectures  [1815 — 17] 
several  attempts  at  the  proof  of  the  Fifth  Postulate,  and 
some  investigations  resembling  those  of  Legendre  have  been 
found. 

However  it  was  only  after  1823  that  he  had  thought  of 
the  Imaginary  Geometry.  This  may  be  inferred  from  the 
manuscript  for  his  book  on  Elementary  Geometry,  where  he 
says  that  we  do  not  possess  any  proof  of  the  Fifth  Postulate, 
but  that  such  a  proof  may  be  possible-^ 


1  For  historical  and  critical  notes  upon  Lobatschewsky  we 
refer  once  and  for  all  to  F.  Engel's  book:  N.  I.  Lobàtschefskij  : 
Zzaci  geo7netrische  Abhandlungen  ans  de?n  Russischen  ubersetzt  tiitf 
Anmerktoigen  und  mit  einer  Biographic  dcs  Verfassers,  (Leipzig, 
Teubner,  1899). 

2  [This  manuscript  had  been  sent  to  St.  Petersburg  in  1823 
to    be    published.    However    it    was   not   printed,    and    it   was    dis- 


Lobatschewsky's  Works.  ge 

Between  1823  and  1825  Lobatschewsky  had  turned 
his  attention  to  a  geometry  independent  of  Euclid's  hypothe- 
sis. The  first  fruit  of  his  new  studies  is  the  Exposition  suc- 
cincie  des  principes  de  la  geometrie  avec  une  demonstration  ri- 
goureuse  dii  théorcme  des parallcles,  read  on  1 2  [24]  Feb.,  1826, 
to  the  Physical  Mathematical  Section  of  the  University  of 
Kasan.  In  this  "Lecture",  the  manuscript  of  which  has 
not  been  discovered,  Lobatschewsky  explains  the  prin- 
ciples of  a  geometry,  more  general  than  the  ordinary  geo- 
metry, where  two  parallels  to  a  given  line  can  be  drawn 
through  a  point,  and  where  the  sum  of  the  angles  of  a  tri- 
angle is  less  than  two  right  angles  [The  Hypothesis  of  the  Acute 
Angle  of  Saccheri  and  Lambert]. 

Later,  in  1829 — 30,  he  published  a  memoir  On  the  Prin- 
ciples of  Geometry ^'^  containing  the  essential  parts  of  the 
preceding  "Lecture",  and  further  apphcations  of  the  new 
theory  in  analysis.  In  succession  appeared  the  Imaginary 
Geometry  [1835],^  New  Principles  of  Geometry,  with  a  Com- 


covered  in  the  archives  of  the  University  of  Kasan. in  1898.  It 
is  clear  from  some  other  remarks  in  this  work  that  he  had  made 
further  advance  in  the  subject  since  1815 — 17.  He  was  now  con- 
vinced that  all  the  first  attempts  at  a  proof  of  the  Parallel  Postulate 
were  unsuccessful,  and  that  the  assumption  that  the  angles  of  a 
triangle  could  depend  only  on  the  ratio  of  the  sides  and  not  upon 
their  absolute  lengths  was  unjustifiable  (cf.  Engel,  loc.cit.  p.  369 — 70).] 

1  Kasan  Bulletin,  (1829 — 1830).  Geometrical  Works  of  Lobat- 
schewsky (Kasan  1883  —  18S6),  Vol.  I  p.  1  —  67.  German  translation 
by  F.  Engel  p.  i  —  66  of  the  work  referred  to  on  the  previous  page. 

Where  the  titles  are  given  in  English  we  refer  to  works  pub- 
lished in  Russian.  The  Geometrical  Works  of  Lobatschewsky  contain 
two  parts;  the  first,  the  memoirs  originally  published  in  Russian; 
the  second,  those  published  in  French  or  German.  It  will  be  seen 
below  that  of  the  works  in  Vol.  i.  several  translations  are  now 
to  be  had. 

2  The  Scientific  Publications  of  the  University  of  Kasan  (1835). 
Geometrical   Works,    Vol.    I,    p.    71 — 120.      German    translation    by 


86    IV.     The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

plett  Theory  of  Parallels  [1835 — 38]^  the  Applications  of  the 
Lnaginary  Geometry  to  Some  Integrals  [1836]^,  then  the 
Geometrie  Imaginaire  [183  7]  3,  and  in  1840,  a  small  book 
containing  a  summary  of  his  work,  Geometrische  Unter' 
suchungen  zur  Theorie  der  Farallellinien,'^  written  in  German 
and  intended  by  Lobatschewsky  to  call  the  attention  of 
mathemiaticans  to  his  researches.  Finally,  in  1855,  a  year 
before  his  death,  when  he  was  already  blind,  he  dictated  and 
pubHshed  in  Russian  and  French  a  complete  exposition  of  his 
system  of  geometry  under  the  title  :  Pangéométrie  ou  precis 
de  geometrie  fondée  sur  une  theorie  generale  et  rigoureuse  des 
par  alleles,  s 

§  40.  Non-Euclidean  Geometry,  just  as  it  was  conceived 
by  Gauss  and  Schweikart  in  1816,  and  studied  as  an  ab- 

H.  LlEBMANN,  with  Notes.  Abhandlungen  zur  Geschichte  der  Mathe- 
matik,  Bd.  XIX,  p.  3—50  (Leipzig,  Teubner,  1904). 

1  Scientific  Publications  of  the  University  of  Kasan  (1835 — 38). 
Geom.  Works.  Vol.  I:  p.  219 — 486.  German  translation  by  F.  Engel, 
p.  67 — 235  of  his  work  referred  to  on  p.  84.  English  translation 
of  the  Introduction  by  G.  B.  Halsted,  (Austin,  Texas,  1897). 

2  Scientific  Publications  of  the  University  of  Kasan.  (1836). 
Geom.  Works,  Vol.  I,  p.  121 — 2l8.  German  translation  by  H,  LlEB- 
MANN; loc.  cit:  p.  51 — 130. 

3  Crelle's  Journal,  Bd.  XVII,  p.  295—320.  (1837).  Geom. 
Works,  Vol.  II,  p.  581—613. 

4  Berlin  (1840).  Geo7)i.  Works,  Vol.  II,  p.  553—578.  French 
translation  by  J.  Houel  in  Mém.  de  Bourdeaux,  T.  IV.  (1866),  and 
also  va.  Recherches  géomèiriques  sur  la  theorie  des parallèles  {?a.xis,  Her- 
mann, 1900).  English  translation  by  G.  B.  Halsted,  (Austin, 
Texas,  1891).     Facsimile  reprint    (Berlin,  Mayer  and  Muller,  1887). 

5  Collection  of  Memoirs  by  Professors  of  the  Royal  University  of 
Kasan  on  the  ^o*''-  anniversary  of  its  foundation.  Vol.  I,  p.  279 — 340. 
(1856).  Also  in  Geom.  Works,  Vol.  II,  p.  617— 680.  In  Russian,  in 
Scientific  Publications  of  the  University  of  Kasan,  (1855).  Italian 
translation,  by  G.  Battaglini,  in  Giornale  di  Mat.  T.  V.  p.  273—336, 
(1867).  German  translation,  by  H.  Liebmann,  Ostwald's  Klassiker 
der  exakten  Wissenschaften,  Nr,   130  (Leipzig,  1 902). 


Lobatschewsky^s  Theory  of  Parallels.  37 

stract  system  by  Taurinus  in  1826,  became  in  1829 — 30 
a  recognized  part  of  the  general  scientific  inheritance. 

To  describe,  as  shortly  as  possible,  the  method  followed 
by  LoBATSCHEWSKY  in  the  construction  of  the  Imaginary  Geo- 
metry or  Pangeometry,  let  us  glance  at  his  G eovietrische  Unter- 
suchungeii  zur  Theorie  der  ParallellÌ7iien  of  1840. 

In  this  work  Lobatschewsky  states,  first  of  all,  a  group 
of  theorems  independent  of  the  theory  of  parallels.  Then  he 
considers  a  pencil  with  vertex 
A,  and  a  straight  line  BC^  in 
the  plane  of  the  pencil,  but 
not  belonging  to  it.  Let  AD 
be  the  line  of  the  pencil  which 
is  perpendicular  to  BC^  and 
AE  that  perpendicular  to 
AD.  In  the  Euclidean  system 
this  latter  line  is  the  only  line  which  does  not  intersect  BC. 
In  the  geometry  of  Lobatschewsky  there  are  other  lines  of  the 
pencil  through  A  which  do  not  intersect  BC.  The  non-inter- 
secting lines  are  separated  from  the  intersectijig  lines  by  the 
two  hues  h,  k  (see  Fig.  44),  which  in  their  turn  do  not  meet 
BC.  [cf.  Saccheri,  p.  42.]  These  lines,  which  the  author  calls 
parallels,  have  each  a  definite  direction  of  parallelistn.  The 
line  //,  of  the  figure,  is  the  parallel  to  the  right:  k,  to  the  left. 
The  angle  which  the  perpendicular  AD  makes  with  one  of 
the  parallels  is  the  ajtgle  0/  parallelism  for  the  length  AD. 
Lobatschewsky  uses  the  symbol  TT  {a)  to  denote  the  angle 
of  parallelism  corresponding  to  the  length  a.  In  the  ordinary 
geometry,  we  have  TT  {a)  =  ()o°  always.  In  the  geometry  of 
Lobatschewsky,  it  is  a  definite  function  of  a,  tending  to 
90°  as  a  tends  to  zero,  and  to  zero  as  a  increases  without 
limit. 

From  the  definition  of  parallels  the  author  then  deduces 
their  principal  properties: 


2,S     IV.     The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

That  if  AD  is  the  parallel  to  £C  for  the  point  A,  it  is 
the  parallel  to  BC  in  that  direction  for  every  point  on  AD 
[permanency]; 

That  if  AD  is  parallel  to  BC,  then  BC  is  parallel  to  . 
AD  [reciprocity]  : 

That  if  the  lines  (2)  and  (3)  are  parallel  to  (i),  then  (2) 
and  (3)  are  parallel  to  each  other  [transitivity]  [cf.  Gauss, 
p.  72];  and  that 

If  AD  and  BC  are  parallel,  AD  is  asymptotic  to  BC. 

Finally,  the  discussion  of  these  questions  is  preceded  by 
the  theorems  on  the  sum  of  the  angles  of  a  triangle,  the 
same  theorems  as  those  already  given  by  Legendre,  and 
still  earlier  by  Saccheri.  There  can  be  little  doubt  that  Lo- 
BATSCHEWSKY  was  familiar  with  the  work  of  Legendre.^ 

But  the  most  important  part  of  the  Imaginary  Geometry 
is  the  construction  of  the  formulae  of  trigonometry. 

To  obtain  these,  the  author  introduces  two  new  figures: 
the  Horocycle  [circle  of  infinite  radius,  cf.  Gauss,  p.  74],  and 
the  Horosphere  ^  [the  sphere  of  infinite  radius],  which  in  the 
ordinary  geometry  are  the  straight  line  and  plane,  respect- 
ively. Now  on  the  Horosphere,  which  is  made  up  of  00  * 
Horocycles,  there  exists  a  geometry  analogous  to  the 
ordinary  geometry,  in  which  Horocycles  take  the  place  of 
straight  lines.  Thus  Lobatschewsky  obtains  this  first  re- 
markable result: 

The  Euclidean  Geometry  [cf.  Wachter,  p.  63],  and.,  in 
particular,  the  ordinary  plane  trigonometry,  hold  upon  the  Hor- 
osphere. 


1  Cf.  LoBATSCHEWSKv's  Criticism  of  I.egendre's  attempt  to 
obtain  a  proof  of  Euclid's  Postulate  in  his  Nexu  Pnnciples  of  Geometry 
(Engel's  translation,  p.  68). 

2  [Lobatschewsky  uses  the  terms  Grcnzkreis,  Grenzkugel  in 
his  German  work:  courbe-limite,  horicycle,  horisphere,  su7-/ace-limite  in 
his  French  work.] 


The  Horocycle  snd  Horocyclic  Surface.  30 

This  remarkable  property  and  another  relating  to  Co- 
axal Horocydes  [concentric  circles  with  infinite  radius]  are 
employed  by  Lobatschewsky  in  deducing  the  formulae  of 
the  new  Plane  and  Spherical  Trigonometries  \  The  formulas 
of  spherical  trigonometry  in  the  new  system  are  found  to  be 
exactly  the  same  as  those  of  ordinary  spherical  trigonometry, 
when  the  elements  of  the  triangle  are  measured  in  right- angles. 

§  41.  It  is  well  to  note  the  form  in  which  Lobatschewsky 
expresses  these  results.  In  the  plane  triangle  ABC,  let  the 
sides  be  denoted  by  a^  b,  c,  the  angles  by  A,  B,  C;  and  let 
T7  (a),  TT  (a),  TT  (c)  be  the  angles  of  parallelism  corresponding 
to  the  sides  a,  b,  c.  Then  Lobatschewsky's  fundamental 
formula  is 

,    .  ,  TT  /7\  TT  /  \         fin  T\  (l>)  sin  TT  \c) 

(4)  cos  A  cos  TT  {b)  cos  TT  {c)  + ^^^ =  1. 

^^^  '  sm  IT  [cij 

It  is  easy  to  see  that  this  formula  and  that  of  Taurinus 
[(i),  p.  79]  can  be  transformed  into  each  other. 

To  pass  from  that  of  Taurinus  to  that  of  Lobatschew- 
sky, we  make  use  of  (3)  of  p.  80,  observing  that  the  angle  B, 
which  appears  in  it,  is  TT  {a). 

For  the  converse  step,  it  is  sufficient  to  use  one*  of  Lo- 
batschewsky's results,  namely  : 

TT  (x)  _  ^ 

(5)  tan-^'  =  a.     "^ 

This  is  the  same  as  the  equation  (3)  of  Taurinus,  under 
another  form. 

The  constant  a  which  appears  in  (5)  is  indeterminate. 
It  represents  the  constant  ratio  of  the  arcs  cut  off  two  Coaxal 


I  It  can  be  proved  that  the  formulae  of  Non-Euclidean  Plane 
Trigonometry  can  be  obtained  without  the  •introduction  of  the 
Ho)-ospke7e.  The  only  result  required  is  the  relation  between  the 
arcs  cut  off  two  Horocydes  by  two  of  their  axes  (cf.  p.  90).  Cf. 
H.  LlEBMANN,  Elementare  Ableitutrg  der  nichteuklidiscken  Trigonometrie. 
Ber.  d.  kòn.  Sach.  Ges.  d.  Wiss.,  Math.  Phys.  Klasse,  (1907). 


QO     IV.     The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

Horocycles  by  a  pair  of  axes,  when  the  distance  between 

these  arcs  is  the  unit  of  length. 

[Fig-  45-] 

If  we  choose,  with  Lobatschew- 
SKY,  a  convenient  unit,  we  are  able 
to  take  a  equal  to  e,  the  base  of 
Natural  Logarithms.  If  we  wish, 
on  the  other  hand,  to  bring  Lo- 
'^  ''  '  batschewsky's  results  into  accord 

with  the  log.-spherical geometry  of  Taurinus,  or  the  Non-Eu- 
clidean geometry  of  Gauss,  we  take 


Then  (5)  becomes  x 

.r  U(x)        ~~T 

(5)  tan^-— =  ir 

2  , 

which  is  the  same  as 

(6)  cosh  7-  =  -. — TT-— ,• 

A  sin    1 1  (x) 

This  result  at  once  transforms  Lobatschewsky's  equa- 
tion (4)  into  the  equation  (i)  of  Taurinus. 

It  follows  that: 

T/ie  log.-spherical  geometry  of  Taurinus  is  identical  with 
the  imagiftary  geometry  \_pa?igeometry]  of  Lobatschewsky. 

§  42.  We  add  the  most  remarkable  of  the  results  which 
Lobatschewsky  deduces  from  his  formulae: 

(a)  In  the  case  of  triangles  whose  sides  are  very  small 
[infinitesimal]  we  can  use  the  ordinary  trigonometrical  for- 
mulae as  the  formulae  ol  Imaginary  Trigonometry,  infinitesi- 
mals of  a  higher  order  being  neglected \ 


I  Conversely,  the  assumption  that  the  Euclidean  Geometry 
holds  for  the  infinitesimally  small  can  be  taken  as  the  starting 
point  for  the  development  of  Non-Euclidean  Geometry.  It  is  one 
of  the  most  interesting  discoveries  from  the  recent  examination  of 


Lobatschewsky's  Trigonometry.  gj 

(b)  If  for  a,  b,  c  are  substituted  ia^  ib,  ic,  the  formulae 
of  Imaginary  Trigonometry  are  transformed  into  those  of  or- 
dinary Spherical  Trigonometry.^ 

(c)  If  we  introduce  a  system  of  coordinates  in  two  and 
three  dimensions  similar  to  the  ordinary  Cartesian  coordinates, 
we  can  find  the  lengths  of  curves,  the  areas  of  surface^-  and 
the  volumes  of  solids  by  the  methods  of  analytical  geometry. 

§  43.  How  was  LoBATSCHEWSKY  led  to  investigate  the 
theory  of  parallels  and  to  discover  the  Imaginary  Geometry? 

We  have  already  remarked  that  Bartels,  Lobatschew- 
sky's teacher  at  Kasan,  was  a  friend  of  Gauss  [p.  84].  If  we 
now  add  that  he  and  Gauss  were  at  Brunswick  together  dur- 
ing the  two  years  which  preceded  his  call  to  Kasan  [1807], 
and  that  later  he  kept  up  a  correspondence  with  Gauss,  the 
hypothesis  at  once  presents  itself  that  they  were  not  without 
their  influence  upon  Lobatschewsky's  work. 

We  have  also  seen  that  before  1807  Gauss  had  attempted 
to  solve  the  problem  of  parallels,  and  that  his  efforts  up  till 
that  date  had  not  borne  other  fruit  than  the  hope  of  overcom- 
ing the  obstacles  to  which  his  researches  had  led  him.  Thus 
anything  that  Bartels  could  have  learned  from  Gauss  before 
1807  would  be  of  a  negative  character.    As  regards  Gauss's 


Gauss's  MSS.  that  the  Princeps  mathematicorum  had  already  fol- 
lowed this  path.     Cf.  Gauss,  Werke,  Bd.  VIII,  p.  255—264. 

Both  the  works  of  Flye  St.  Marie,  [Thhrie  analytlque  sur  la 
thèorie  des  parallèies,  (Paris,  1871)],  and  of  KILLING  [Die  7iichteuklid- 
ischen  Raiwiformen  in  analytischer  Behandlung,  (Leipzig,  1881)],  are 
founded  upon  this  principle.  In  addition,  the  formulae  of  trigono- 
metry have  been  obtained  in  a  simple  manner  by  the  application 
of  the  same  principle,  and  the  use  of  a  few  fundamental  ideas,  by 
M.  Simon.  [Cf.  M.  Simon,  Die  Trigonometrie  in  der  absoluten  Geotnetrie, 
Crelle's  Journal,  Bd.  109,  p.  187 — 198  (1892)]. 

I  This  result  justifies  the  method  followed  by  Taurinus  in 
the  construction  of  his  log. -spherical  geometry. 


Q2     IV.     The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

later  views,  it  appears  quite  certain  that  Bartels  had  no  news 
of  them^  so  that  we  can  be  sure  that  Lobatschewsky  created 
his  geometry  quite  independently  of  any  influence  from  Gauss.* 
Other  influences  might  be  mentioned:  e.  g.,  besides  Legendre, 
the  works  of  Saccheri  and  Lambert,  which  the  Russian  geo- 
meter might  have  known,  either  directly  or  through  Klugel 
and  MoNTUCLA.  But  we  can  come  to  no  definite  decision 
upon  this  question^.  In  any  case,  the  failure  of  the  demon- 
strations of  his  predecessors,  or  the  uselessness  of  his  own 
earlier  researches  [1815 — 17],  induced  Lobatschewsky,  as 
formerly  Gauss,  to  believe  that  the  difficulties  which  had 
to  be  overcome  were  due  to  other  causes  than  those  to 
which  until  then  they  had  been  attributed.  Lobatschewsky 
expresses  this  .thought  clearly  in  the  Nau  Principles  of 
Geometry  of  1825,  where  he  says: 

'The  fruitlessness  of  the  attempts  made,  since  Euclid's 
time,  for  the  space  of  2000  years,  aroused  in  me  the  suspicion 
that  the  truth,  which  it  was  desired  to  prove,  was  not  contained 
in  the  data  themselves;  that  to  establish  it  the  aid  of  experi- 
ment would  be  needed,  for  example,  of  astronomical  obser- 
vations, as  in  the  case  of  other  laws  of  nature.  When  I  had 
finally  convinced  myself  of  the  justice  of  my  conjecture  and 
beheved  that  I  had  completely  solved  this  difficult  question, 
"^I  wrote,  in  1826,  a  memoir  on  this  subject  {Exposition  suc- 
cincte  des  principes  de  la  Géomctrie\.'  ^ 

The  words  of  Lobatschewsky  afford  evidence  of  a  phil- 
osophical conception  of  space,  opposed  to  that  of  Kant, 
which  was  then  generally  accepted.  The  Kantian  doctrine 
considered  space  as  a  subjective  intuition,  a  necessary  presup- 
position of  every  experience.  Lobatschewsky's  doctrine  was 


1  Cf.  the  work  of  F.  Engel,   quoted    on  p.  84.     Zweiter  Teil; 
Lobatschefskijs  Leben  unci  Schriftett.     Cap.  VI,  p.  373 — 383. 

2  Cf.  Segre's  work,  quoted  on  p.  44. 

3  Cf.  p.  67  of  Engel's  work  named  above. 


The  Pangeometry.  q-ì 

rather  allied  to  sensualism  and  the  current  empiricism,  and 
compelled  geometry  to  take  its  place  again  among  the  ex- 
perimental sciences.^ 

§44.  It  now  remains  to  describe  the  relation  of  Lobat- 
scHE\vsK"S''s  Paiigeo7netry  to  the  debated  question  of  the  Eu- 
clidean Postulate.  This  discussion,  as  we  have  seen,  aimed 
at  constructing  the  Theory  of  Parallels  with  the  help  of  the 
first  28  propositions  of  Euclid. 

So  far  as  regards  this  problem,  Lobatschewsky,  having 
defined  parallelism,  assigns  to  it  the  distinguishing  features 
of  reciprocity  and  transitivity.  The  property  of  equidistance 
then  presents  itself  to  Lobatschewsky  in  its  true  light.  Far 
from  being  indissolubly  bound  up  with  the  first  28  proposit- 
ions of  Euclid,  it  contains  an  element  entirely  new. 

The  truth  of  this  statement  follows  directly  from  the  ex- 
istence of  the  Pangeometry  [a  logical  deductive  science  founded 
upon  the  said  28  propositions  and  on  the  negation  of  the 
Fifth  Postulate],  in  which  parallels  are  not  equidistatit,  but  are 
asymptotic.  Further,  we  can  be  sure  that  the  Pangeometry 
is  a  science  in  which  the  results  follow  logically  one  from  the 
other,  i.  e.,  are  free  from  internal  contradictions.  To  prove 
this  we  need  only  consider,  with  Lobatschewsky,  the  analyt- 
ical form  in  which  it  can  be  expressed. 

This  point  is  put  by  Lobatschewsky  toward  the  end  of 
his  work  in  the  following  way: 

'Now  that  we  have  shown,  in  what  precedes,  the  way  in 
which  the  lengths  of  curves,  and  the  surfaces  and  volumes  of 
solids  can  be  calculated,  we  are  able  to  assert  that  the  Pan- 
geometry  is  a  complete  system  of  geometry.  A  single  glance 


I  Cf.  The  discourse  on  Lobatschewsky  by  A.  Vasiliev, 
(Kasan,  1893).  German  translation  by  Engel  in  Schlomilch's  Zeit- 
schrift,  Bd.  XI,  p.  205 — 244  (1895).  'English  translation  by  Halsted, 
(Austin,  Texas,  1 895). 


94 


IV.     The  Founders  of  Non-Euclidean  Geometry  (Contd. 


at  the  equations  which  express  the  relations  existing  between 
the  sides  and  angles  of  plane  triangles,  is  sufficient  to  show 
that,  setting  out  from  them,  Pangeometry  becomes  a  branch  of 
analysis,  including  and  extending  the  analytical  methods  of 
ordinary  geometry.  We  could  begin  the  exposition  of  Pan- 
geometry  with  these  equations.  We  could  then  attempt  to 
substitute  for  these  equations  others  which  would  express  the 
relations  between  the  sides  and  angles  of  every  plane  triangle. 
However,  in  this  last  case,  it  would  be  necessary  to  show 
that  these  new  equations  were  in  accord  with  the  fundamental 
notions  of  geometry.  The  standard  equations,  having  been 
deduced  from  these  fundamental  notions,  must  necessarily  be 
in  accord  with  them,  and  all  the  equations  which  we  would 
substitute  for  them,  if  they  cannot  be  deduced  from  the  equa- 
tions, would  lead  to  results  contradicting  these  notions.  Our 
equations  are,  therefore,  the  foundation  of  the  most  general 
geometry,  since  they  do  not  depend  on  the  assumption  that 
the  sum  of  the  angles  of  a  plane  triangle  is  equal  to  two  right 
angles.'  ' 

§  45.  To  obtain  fuller  knowledge  of 
the  nature  of  the  constant  k  contained  im- 
plicity  in  Lobatschewsky's  formulae,  and 
exphcitly  in  those  of  Taurinus,  we  must 
apply  the  new  trigonometry  to  some  actual 
case.  To  this  end  Lobatschewsky  used  a 
triangle  ABC,  in  which  the  side  BC  {a)  is 
equal  to  the  radius  of  the  earth's  orbit, 
and  ^  is  a  fixed  star,  whose  direction  is 
perpendicular  to  BC  (Fig.  46).  Denote 
hy  2  p  the  maximum  parallax  of  the  star 
A.    Then  we  have 


1  Cf.  the  Italian  translation  of  the  Pangéomélrie,  Giornale  di 
Mat,,  T.  V.  p.  334;  or  p.  75  of  the  German  translation  referred  to 
on  p.  86. 


Astronomy  and  Lobatschewsky's  Theory.  ge 

Therefore 

I  /it  \  I  —  tan/ 

tan  -T](a)>  tan   (--  -/j  =  .^^^^ 

a 
But  tan  l-  T\  (a)  =  e      J  [cf.  p.  90], 

a 

Therefore       .'^<i^'-"^. 
^  I  —  tan/ 

IT 

But  on  the  hypothesis  /  <C      j  we  have 

Also,   tan  2/  =      2 tan/ 
I  —  tan2/ 
=  2  (tan/  +  tan3/  +  tan^/  +  ...). 

Therefore  we  have 

-^  <  tan  2/. 

Take  now,  with  Lobatschewsky,  the  parallax  of  Sirius 
as   i",  24. 

From  the  value  of  tan  2  /,  we  have 

—  <C  0,000006012. 

This  result  does  not  allow  us  to  assign  a  value  to  k, 
but  it  tells  us  that  it  is  very  great  compared  with  the  diam- 
eter of  the  earth's  orbit.  We  could  repeat  the  calculation 
for  much  smaller  parallaxes,  for  example  o",i,  and  we 
would  find  k  to  be  greater  than  a  million  times  the  diameter 
of  the  earth's  orbit. 

Thus,  if  the  EucUdean  Geometry  and  the  Fifth  Postul- 
ate are  to  hold  in  actual  space,  k  must  be  infinitely  great. 
That  is  to  say,  there  must  be  stars  whose  parallaxes  are  in- 
definitely small. 

However  it  is  evident  that  we  can  never  state  whether 
this  is  the  case  or  not,  since  astronomical  observations  will 


q5     IV.     The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

always  be  true  only  within  certain  limits,  Yet^  knowing  the 
enormous  size  of  k  in  comparison  with  measurable  lengths, 
we  must,  with  Lobatschewsky,  admit  that  the  Euclidean 
hypothesis  is  valid  for  all  practical  purposes. 

We  would  reach  the  same  conclusion  if  we  regarded 
the  question  from  the  standpoint  of  the  sum  of  the  angles  of 
a  triangle.  The  results  of  astronomical  observations  show  that 
the  defect  of  a  triangle,  whose  sides  approach  the  distance 
of  the  earth  from  the  sun,  cannot  be  more  than  o",ooo3. 
Let  us  now  consider,  instead  of  an  astronomical  triangle,  one 
drawn  on  the  Earth's  surface,  the  angles  of  which  can  be 
directly  measured.  In  consequence  of  the  fundamental  theorem 
that  the  area  of  a  triangle  is  proportional  to  its  defect,  the 
possible  defect  would  fall  within  the  limits  of  experimental 
error.  Thus  we  can  regard  the  defect  as  zero  in  experimental 
work,  and  Euclid's  Postulate  will  hold  in  the  domain  of  ex- 
perience.^ 

Johann  Bolyai  [1802 — 1860]. 

§  46.  J.  Bolyai  a  Hungarian  officer  in  the  Austrian 
army,  and  son  of  Wolfgang  Bolyai,  shares  with  Lobat- 
schewsky the  honour  of  the  discovery  of  Non-Euclidean  geo- 
metry. From  boyhood  he  showed  a  remarkable  aptitude  for 
mathematics,  in  which  his  father  himself  instructed  him.  The 
teaching  of  Wolfgang  quickly  drew  Johann's  attention  to 
Axiom  XL  To  its  demonstration  he  set  himself,  in  spite  of 
the  advice  of  his  father,  who  sought  to  dissuade  him  from  > 
the  attempt.  In  this  way  the  theory  of  parallels  formed  the 
favourite  occupation  of  the  young  mathematician,  during  his . 
course  [1817 — 22]  in  the  Royal  College  for  Engineers  at 
Vienna. 


I  For  the  contents  of  this  section,  cf.  Lobatschewsky,  On 
the  Principles  of  Geometry,  See  p.  22 — 24  of  Engel's  work  named 
on  p.  84.     Also  Engel's  remarks  on  p.  24S — 252  of  the  same  work. 


Johann  Boiyai's  Earlier  Work. 


97 


At  this  time  Johann  was  an  intimate  friend  of  Carl 
SzAsz  [1798-185 3]  and  the  seeds  of  some  of  the  ideas,  which 
led  BoLYAi  to  create  the  Absolute  Science  of  Space,  were  sown 
in  the  conversations  of  the  two  eager  students. 

It  appears  that  to  Szasz  is  due  the  distinct  idea  of  con- 
sidering the  parallel  through  £  to  the  line  AM  as  the  limit- 
ing position  of  a  secant  BC  turning  in  a  definite  direction 
about  JB;  that  is,  the  idea  of  consid- 
ering BC  as  parallel  to  AM,  when 
BC,  in  the  language  of  Szasz,  de- 
taches itself  ^springs  away)  from  AM 
(Fig.  47).  To  this  parallel  Bolyai 
gave  the  name  of  asymptotic  parallel 
or  asymptote,  [cf  Saccheri].  From 
the  conversations  of  the  two  friends 
were  also  derived  the  conception  of 
the  line  equidistant  from  a  straight  line, 
and  the  other  most  important  idea  of 
the  Paracycle  {lÌ7ìiiting  curve  or  horo- 
ry*;/.?  of  Lobatschewsky).  Further  they 
recognised  that  the  proof  of  Axiom  XI  would  be  obtained 
if  it  could  be  shown  that  the  Paracycle  is  a  straight  line. 

When  Szasz  left  Vienna  in  the  beginning  of  1821  to 
undertake  the  teaching  of  Law  at  the  College  of  Nagy-Enyed 
(Hungary),  Johann  remained  to  carry  on  his  speculations 
alone.  Up  till  1820  he  was  filled  with  the  idea  of  finding 
a  proof  of  Axiom  XI,  following  a  path  similar  to  that  of 
Saccheri  and  Lambert.  Indeed  his  correspondence  with 
his  father  shows  that  he  thought  he  had  been  successful  in 
his  aim. 

The  recognition  of  the  mistakes  he  had  made  was  the 
cause  of  Johann's  decisive  step  towards  his  future  discoveries, 
since  he  realised  'that  one  must  do  no  violence  to  nature, 
nor  model  it  in  conformity  to  any  blindly  formed  chimsera; 

7 


q3        IV.    The  Founders  of  Non-Euclidean  Geometry  (^Contd.) 

that,  on  the  other  hand,  one  must  reguard  nature  reasonably 
and  naturally,  as  one  would  the  truth,  and  be  contented  only 
with  a  representation  of  it  which  errs  to  the  smallest  possible 
extent.' 

Johann  Bolyai,  then,  set  himself  to  construct  an  abso- 
hite  theory  of  space,  following  the  classical  methods  of  the 
Greeks:  that  is,  keeping  the  deductive  method^  but  without 
deciding  a  priori  on  the  truth  or  error  of  the  FifthPostulate. 

§  47.  As  early  as  1823  Bolyai  had  grasped  the  real 
nature  of  his  problem.  His  later  additions  only  concerned 
the  material  and  its  formal  expression.  At  that  date  he  had 
discovered  the  formula: 

a 
e       ■^   =  tan  -  -  , 

connecting  the  angle  of  parallelism  with  the  line  to  which  it 
corresponds  [cf.  Lobatschewsky,  p.  89].  This  equation  is 
the  key  to  all  Non- Euclidean  Trigonometry.  To  illustrate  the 
discoveries  which  Johann  made  in  this  period,  we  quote  the 
following  extract  from  a  letter  which  he  wrote  from  Temesvar 
to  his  father,  on  Nov.  3,  1823:  'I  have  now  resolved  to  pub- 
lish a  work  on  the  theory  of  parallels,  as  soon  as  I  shall  have 
put  the  material  in  order,  and  my  circumstances  allow  it.  I 
have  not  yet  completed  this  work,  but  the  road  which  I  have 
followed  has  made  it  almost  certain  that  the  goal  will  be 
attained,  if  that  is  at  all  possible:  the  goal  is  not  yet  reached, 
but  I  have  made  such  wonderful  discoveries  that  I  have  been 
almost  overwhelmed  by  them,  and  it  would  be  the  cause  of 
continual  regret  if  they  were  lost.  When  you  will  see  them, 
you  too  will  recognize  it.  In  the  meantime  I  can  say  only 
this  :  /  have  created  a  ne^v  universe  from  nothing.  All  that  I 
have  sent  iyou  till  now  is  but  a  house  of  cards  compared  to 
the  tower.  I  am  as  fully  persuaded  that  it  will  bring  me 
honour,  as  if  I  had  already  completed  the  discovery.' 


J.  Bolyai's  Theory  of  Parallels.  gg 

Wolfgang  expressed  the  wish  at  once  to  add  his  son's 
theory  to  the  Tentatnen  since  'if  you  have  really  succeeded 
in  the  question,  it  is  right  that  no  time  be  lost  in  making  it 
public,  for  two  reasons:  first,  because  ideas  pass  easily  from 
one  to  another,  who  can  anticipate  its  publication;  and  se- 
condly, there  is  some  truth  in  this,  that  many  things  have  an 
epoch,  in  which  they  are  found  at  the  same  time  in  several 
places,  just  as  the  violets  appear  on  every  side  in  spring. 
Also  every  scientific  struggle  is  just  a  serious  war,  in  which 
I  cannot  say  when  peace  will  arrive.  Thus  we  ought  to 
conquer  when  we  are  able,  since  the  advantage  is  always  to 
the  first  comer.' 

Little  did  Wolfgang  Bolyai  think  that  his  presentiment 
would  correspond  to  an  actual  fact  (that  is,  to  the  simulta- 
neous discovery  of  Non-Euclidean  Geometry  by  the  work  of 
Gauss,  Taurinus,  and  Lobatschewsky). 

In  1825  Johann  sent  an  abstract  of  his  work,  among 
others,  to  his  father  and  to  J.Walter  von  Eckwehr  [1789 — 
1857],  his  old  Professor  at  the  Military  School.  Also  in  1829 
he  sent  his  manuscript  to  his  father.  Wolfgang  was  not 
completely  satisfied  with  it,  chiefly  because  he  could  not  see 
why  an  indeterminate  constant  should  enter  into  Johann's 
formulae.  None  the  less  father  and  son  were  agreed  in 
pubhshing  the  new  theory  of  space  as  an  appendix  to  the 
first  volume  of  the  Tentamen: — 

The  title  of  Johann  Bolyai's  work  is  as  follows. 

Appendix  scientiam  spatii  absolute  veram  exhibens:  a 
ventate  aut  falsitate  Axiomatis  XI.  Euclidei,  a  priori  haud 
unquam  decidenda,  independentem  :  adjecta  ad  casum  falsitaiis 
quadratura  circuii  geometrica.^ 


I  A  reprint — Edition  de  Luxe — was  issued  by  the  Hungarian 
Academy  of  Sciences,  on  the  occasion  of  the  first  centenary  of 
the    birth    of   the    author    (Budapest,  1902).     See    also  the  English 

7* 


lOO     IV.     The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

The  Appendix  was  sent  for  the  first  time  [June,  1831] 
to  Gauss,  but  did  not  reach  its  destination;  and  a  second 
time,  in  January,  1832.  Seven  weeks  later  (March  6,  1832), 
Gauss  replied  to  Wolfgang  thus: 

"If  I  commenced  by  saying  that  I  am  unable  to  praise 
this  work  (by  Johann),  you  would  certainly  be  surprised 
for  a  moment.  But  I  cannot  say  otherwise.  To  praise  it, 
would  be  to  praise  myself.  Indeed  the  whole  contents  of 
the  work,  the  path  taken  by  your  son,  the  results  to  which  he 
is  led,  coincide  almost  entirely  with  my  meditations,  which 
have  occupied  my  mind  partly  for  the  last  thirty  or  thirty- 
five  years.  So  I  remained  quite  stupefied.  So  far  as  my 
own  work  is  concerned,  of  which  up  till  now  1  have  put  little 
on  paper,  my  intention  was  not  to  let  it  be  published  during 
my  lifetime.  Indeed  the  majority  of  people  have  not  clear 
ideas  upon  the  questions  of  which  we  are  speaking,  and  I 
have  found  very  few  people  who  could  regard  with  any  special 
interest  what  I  communicated  to  them  on  this  subject.  To 
be  able  to  take  such  an  interest  it  is  first  of  all  necessary 
to  have  devoted  careful  thought  to  the  real  nature  of  what  is 
wanted  and  upon  this  matter  almost  all  are  most  uncertain. 
On  the  other  hand  it  was  my  idea  to  write  down  all  this  later 
so  that  at  least  it  should  not  perish  with  me.  It  is  therefore  a 
pleasant  surprise  for  me  that  I  am  spared  this  trouble,  and  I 
am  very  glad  that  it  is  just  the  son  of  my  old  friend,  who 
takes  the  precedence  of  me  in  such  a  remarkable  manner." 

Wolfgang  communicated  this  letter  to  his  son,  adding: 
"Gauss's   answer  with  regard  to   your   work  is  very  satis- 

translation  by  Halsted,  T/ie  Science  Absolute  of  Space,  (Austin,  Texas^ 
1896).  An  Italian  translation  by  G.  B.  Battagmni  appeared  in  the 
Giornale  di  Mat.,  T.  VI,  p- 97  — 115  (1868).  Also  a  French  trans- 
lation by  HOUEL,  in  Mém.  de  la  Soc  des  Se.  de  Bordeaux,  T.  V- 
p.  189 — 248  (1867).  Cf.  also  Frischauf,  Absoluie  Geometrie  nach 
Johann  Bolyai,  (Leipzig,  Teubner,  1872). 


Gauss's  Praise  of  Eolyai's  Work.  lOI 

factory  and  redounds  to  the  honour  of  our  country  and  of 
our  nation." 

Altogether  different  was  the  effect  Gauss's  letter  pro- 
duced on  Johann.  He  was  both  unable  and  unwilling  to 
convince  himself  that  others,  earlier  than  and  independent  of 
him,  had  arrived  at  the  No7i- Euclidean  Geometry.  Further  he 
suspected  that  his  father  had  communicated  his  discoveries 
to  Gauss  before  sending  him  the  Appendix  and  that  the  latter 
wished  to  claim  for  himself  the  priority  of  the  discovery. 
And  although  later  he  had  to  let  himself  be  convinced  that 
such  a  suspicion  was  unfounded,  Johann  always  regarded 
the  "Prince  of  Geometers"  with  an  unjustifiable  aversion.  * 

§  48.  We  now  give  a  short  description  of  the  most 
important  results  contained  in  Johann  Bolyai's  work: 

a)  The  definition  of  parallels  and  their  properties  in- 
dependent of  the  Euchdean  postulate. 

b)  The  circle  and  sphere  of  infinite  radius.  The  geo- 
metry on  the  sphere  of  infinite  radius  is  identical  with  ordi- 
nary plane  geometry. 

c)  Spherical  Trigonometry  is  independent  of  Euclid's 
Postulate.   Direct  demonstration  of  the  formulae. 

d)  Plane  Trigonometry  in  Non-Euclidean  Geometry. 
Applications  to  the  calculation  of  areas  and  volumes. 

e)  Problems  which  can  be  solved  by  elementary  me- 
thods. Squaring  the  circle,  on  the  hypothesis  that  the  Fifth 
Postulate  is  false. 

While  LoBATSCHEWSKY  has  given  the  Imaginary  Geo- 
metry a  fuller  development  especially  on  its  analytical  side, 

I  For  the  contents  of  this  and  the  preceding  article  seeSxAcKEL, 
Die  Entdeckung  der  7iichteuklidischeii  Geometrie  durch  jfohatin  Bolyai. 
Math.  u.  Naturw.  Berichte  aus  Ungarn.     Bd.  XVII,  [1901]. 

Also  StAckel  u.  Engel.  Gauss,  die  beiden  Bolyai  und  die 
nichteuklidische  Geometrie.  Math.  Ann.  Bd.  XLIX,  p.  149 — 167  [1897]. 
Bull.  So.  Math.  (2)  T.  XXI,  pp.  206—228  [1897]. 


I02     IV.    The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

BoLYAi  entered  more  fully  into  the  question  of  the  depen- 
dence or  independence  of  the  theorems  of  geometry  upon 
Euclid's  Postulate.  Also  while  Lobatschewky  chiefly  sought 
to  construct  a  system  of  geometry  on  the  negation  of  the 
said  postulate,  Johann  Bolyai  brought  to  light  the  pro- 
positions and  constructions  in  ordinary  geometry  which  are 
independent  of  it.  Such  propositions,  which  he  calls  ab- 
solutely true,  pertain  to  the  absolute  science  of  space.  We 
could  find  the  propositions  of  this  science  by  comparing 
EucUd's  Geometry  with  that  of  Lobatschewsky.  Whatever 
they  have  in  common,  e.  g.  the  formulae  of  Spherical  Trigon- 
ometry, pertains  to  the  Absolute  Geometry.  Johann  Bolyai, 
however,  does  not  follow  this  path.  He  shows  directly,  that 
is  independently  of  the  Euchdean  Postulate,  that  his  propos- 
itions are  absolutely  true. 

§  49.  One  of  BoLYAi's  absolute  theorems,  remarkable 
for  its  simplicity  and  neatness,  is  the  following: 

The  sines  of  the  angles  of  a  rectilinear  triangle  are  to  one 
another  as  the  circumferences  of  the  circles  whose  radii  are 
equal  to  the  opposite  sides. 


A 


A  ^ 

b 


B' 

B 

Fig.  48. 

Let  ABC  be  a  triangle  in  which  C  is  a  right  angle,  and 
BB'  the  perpendicular  through  B  to  the  plane  of  the  triangle. 

Draw  the  parallels  through  A  and  C  to  BB'  in  the 
same  sense. 

Then  let  the  Horosphere  be  drawn  through  A  (eventually 
the  plane)  cutting  the  lines  AA\  BB'  and  CC,  respectively, 
in  the  points  A,  M,  and  N. 


Bolyai's  Theorem.  -  10^ 

If  we  denote  by  a\  b\  c  the  sides  of  the  rectangular 
triangle  AAIN  on  the  Horosphere,  it  follows  from  what  has 
been  said  above  [cf.  §  48  (b)]  that 

sin  AMN  =  — . 

But  two  arcs  of  Horocycles  on  the  Horosphere  are  pro- 
portional to  the  circumferences  of  the  circles  which  have 
these  arcs  for  their  (horocyclic)  radii. 

If  we  denote  by  circumf.  x  the  circumference  of  the 
circle  whose  (horocyclic)  radius  is  x',  we  can  write: 

A  T^jT^T         circumf.  b' 

Sin  AMN  =  -. J—,. 

circumf.  c 

On  the  other  hand,  the  circle  traced  on  the  Horosphere 
with  horocyclic  radius  of  length  x\  can  be  regarded  as  the 
circumference  of  an  odinary  circle  whose  radius  (rectilinear) 
is  half  of  the  chord  of  the  arc  2  x'  of  the  Horocycle. 

Denoting  by  O  •^  the  circumference  of  the  circle  whose 
(rectilinear)  radius  is  x,  and  observing  that  the  angles  ABC 
and  AMN  are  equal,  the  preceding  equation  taken  from 

sin  ABC  =  -^. 

From  the  property  of  the  right  angled  triangle  ABC 
expressed  by  this  equation,  we  can  deduce  Bolyai's  theorem 
enunciated  above,  just  as  from  the  Euclidean  equation 

sin  ABC  =  — 

c 

we  can  deduce  that  the  sines  of  the  angles  of  a  triangle  are 
proportional  to  the  opposite  sides.    {Appendix  §  25.] 

Bolyai's  Theorem  may  be  put  shortly  thus: 
(i)  O^  '•  O^  '•  O^  =  sin  ^  :  sin  B  :  sin  C. 

If  we  wish  to  discuss  the  geometrical  systems  separately 
we  will  have 

(i)     In  the  case  of  the  Euclidean  Hypothesis, 

O-^   =    2  TTAT. 


I04      ^^'  ^^^  Founders  of  Non-Euclidean  Geometry  (Contd.). 


(!') 


2  11;^  sinh  -r* 


(i  ")       sinh  —  :  sinh  ^  :  sinh  -r  ==  sin  A  :  sin  B  :  sin  C. 


Thus,  substituting  in  (i),  we  have 

a:b'.C'.  ==  sin  ^  :  sin  ^  :  sin  C. 
(ii)       In  the  case  of  the  Non-Eudidean  Hypothesis, 

Then  substituting  in  (i)  we  have 

b 

k  ' T  •  " /• 

This  last  relation  may  be  called  the  Sine  Theorem  of  the 
Bolyai-Lobatschewsky  Geometry. 

From  the  formula  (i)  Bolyai  deduces,  in  much  the 
same  way  as  the  usual  relations  are  obtained  from  (i),  the 
proportionality  of  the  sines  of  the  angles  and  the  opposite  sides 
in  a  spherical  triangle.  From  this  it  follows  that  Spherical 
Trigonometry  is  independent  of  the  Euclidean  Postulate 
{Appendix  8  26]. 

This  fact  makes  the  importance  of  Bolyai's  Theorem 
still  clearer. 


§  50.  The  following  construction  for  a  parallel  through 
the  point  Z>  to  the  straight  line  ^iV  belongs  also  to  the  Ab- 
solute Geometry  [Appendix  %  34]. 

Draw  the  perpendiculars  DB  and  AE  to  AN  [Fig.  49]. 
fi  D 


Fig.  49. 


Also  the  perpendicular  DE  to   the  line  AE.    The   angle 
EDB  of  the  quadrilateral  ABDE,  in  which  three  angles 


Bolyai's  Parallel  Construction.  IO5 

are  right  angles,  is  a  right  angle  or  an  acute  angle,  according 
as  ED  is  equal  to  or  greater  than  AB. 

With  centre  A  describe  a  circle  whose  radius  is  equal 
to  ED. 

It  will  intersect  DB  at  a  point  (9,  coincident  with  B  or 
situated  between  B  and  D. 

The  angle  which  the  line  AO  juakes  with  DB  is  the 
angle  of  parallelisin  corresponding  to  the  segmefit  BD.^ 
[Appendix  §27.] 

Therefore  a  parallel  to  AN  through  D  can  be  con- 
structed by  drawing  the  line  DM  so  that  <C  BDM  is 
equal  to  <^  AOB.^ 


1  We  give  a  sketch  of  Bolyai's  proof  of  this  theorem:  The 
circumferences  of  the  circles  with  radii  AB  and  ED,  traced  out 
by  the  points  B  and  D  in  their  rotation  about  the  line  AE,  can 
be  considered  as  belonging,  the  first  to  the  plane  through  A  per- 
pendicular to  the  axis  AE,  the  second  to  an  Equidistant  Surface 
for  this  plane.  The  constant  distance  between  the  surface  and 
the  plane  is  the  segment  BD^^d.  The  ratio  between  these  two 
circumferences  is  thus  a  function  of  d  only.  Using  Bolyai's 
Theorem,  S  49>  and  applying  it  to  the  two  rightangled  triangles 
ADE  and  ADB,  this  ratio  can  be  expressed  as 

O  AB  :  O  ED  =  sin  u  :  sin  v. 
From  this  it  is  clear  that  the  ratio  sin  k  :  sin  v  does  not  vary  if 
the  line  AE  changes  its  position,  remaining  always  perpendicular 
to  AB,  while  d  remains  fixed.  In  particular,  if  the  foot  of  AE 
tends  to  infinity  along  AJV,  it  tends  to  TT  {d)  and  z/  to  a  right  angle. 
Consequently, 

QAB  ■.QED  =  sm  T\{d):\. 

On    the    other    hand    in    the    right-angled  triangle  AOB,    we  have 
the  equation 

QAB:QiAO  =  sin  AOB  :  i. 
This,    with    the    preceding   equation,   is  sufficient    to   establish  the 
equality  of  the  angles  TT  {d)  und  AOB. 

2  Cf.  Appendix  III  to  this  volume. 


I06     IV.    The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

§  51.  The  most  interesting  of  the  Non-Euclidean  con- 
structions given  by  Bolvai  is  that  for  the  squaring  of  the 
circle.  Without  keeping  strictly  to  Bolyai's  method,  we  shall 
explain  the  principal  features  of  his  construction. 

But  we  first  insert  the  converse  of  the  construction  of 
§  50,  which  is  necessary  for  our  purpose. 

On  the  Non-Euclidea7i  Hypothesis  to  draw  the  segment 
which  corresponds  to  a  given  {acute)  angle  of  parallelism. 

Assuming  that  the  theorem,  that  the  three  perpendiculars 
from  the  angular  points  of  a  triangle  on  the  opposite  sides 
intersect  eventually,  is  also  true  in  the  Geometry  of  Bolyai- 
LoBATSCHEWSKY,  on  the  line  AB  bounding  the  acute  angle 
BAA'  take  a  point  B,  such  that  the  parallel  BB'  to  AA 
through  B  makes  an  acute  angle  {ABB')  with  AB.    [Fig.  50.] 


Fig.  50. 


The  two  rays  AA ,  BB',  and  the  line  AB  may  be 
regarded  as  the  three  sides  of  a  triangle  of  which  one  angular 
point  is  Coo  )  common  to  the  two  parallels  AA,  BB'.  Then 
the  perpendiculars  from  A,  B,  to  the  opposite  sides,  meet  in 
he  point  O  inside  the  triangle,  and  the  perpendicular  from 
Coo  to  AB  also  passes  through  O. 

Thus,  if  the  perpendicular  OL  is  drawn  from  O  to  AB, 
the  segment  AL  will  have  been  found  which  corresponds  to 
the  angle  of  parallelism  BAA . 


Bolyai's  Parallel  Construction  (Contd.). 


107 


As  a  particular  case  the  angle  BAA'  could  be  45°. 
Then  AL  would  be  Schweikart's  Constant  [cf.  p.  76]. 

We  note  that  the  problem  which  we  have  just  solved 
could  be  enunciated  thus: 

To  draw  a  line  which  shall  be  parallel  to  one  of  the  lines 
bounding  an  acute  angle  and  perpendicular  to  the  other.  "^ 

§  52.  We  now  show  how  the  preceding  result  is  used 
to  construct  a  square  equal  in  area  to  the  maximum  triangle. 

The  area  of  a  triangle  being 

k'{M—KA  —  ^B—^C), 
the  maximum  triangle,  i.  e.  that  for  which  the  three  angular 
points  are  at  infinity,  will  have  for  area 

A  =  k^  TT. 

To  find  the  angle  oi  of  a  square  whose  area  is  k'^n,  we  need 
only  remember  (Lambert,  p.  46)  that  the  area  of  a  polygon^ 
as  well  as  of  a  triangle,  is  proportional  to  its  defect.  Thus 
we  have  the  equation 

k^  11  =  k'^  (2  TT  —  4  uj), 
from  which  it  follows  that 


UU  =  "  IT 

4 


45^ 


Assuming  this^  let  us  consider  the 
right-angled  triangle  0AM  (Fig,  51), 
which  is  the  eighth  part  of  the  required 
square.  Putting  OM  =  <?,  and  ap- 
plying the  formula  (2)  of  p,  80  we  \/ 
obtain 


O) 

0 

Ab 

/    2 
/45° 

/ 

a 

M 


F<g.  51- 


cosh  -r  = 

k 


or  cosh  -r  = 
k 


cos    22"  30 

sin  45" 

sin  670  30' 

sin  450 


cated. 


I  Bolyai's  solution  [Appendix.,  §  35]  is,  however,  more  compii- 


I08      IV.    The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

If  we  now  draw,  as  in  §  51,  the  two  segments  b' ,  c, 
which  correspond  to  the  angles  67°  30'  and  45°,  and  if  we 
remember  that  [cf.  p.  90  (6)] 

cosh  -T-  =   ^-TfT-Tj 
k  smTT  (;>:)' 

the  following  relation  must  hold  between  a,  b'  and  c, 
cosh  -J  cosh  -J  =  cosh  -j. 

Finally  if  we  take  b'  as  side,  and  /  as  hypotenuse  of  a  right- 
angled  triangle,  the  other  side  of  this  triangle,  by  formula  (i) 
of  p.  7  9,  is  determined  by  the  equation 

cosh  -,~  cosh  -,-  =  cosh  -r-. 

Then  comparing  these  two  questions,  we  obtain 

a   =  a. 
Constructing  a  in  this  way,  we  can  immediately  find  the 
square  whose  area  is  equal  to  that  of  the  maximum  triangle. 

§  53.  To  construct  a  circle  whose  area  shall  be  equal 
to  that  of  this  square,  that  is,  to  the  area  of  the  maximum 
triangle,  we  must  transform  the  expression  for  the  area  of 
a  circle  of  radius  r 

2  TT  /C'^  (  cosh  -, I  ) , 

given  on  p.  81,  by  the  introduction  of  the  angle  of  parallelism 
TT( — j,  corresponding  to  half  the  radius. 

Then  we  have'  for  the  area  of  this  circle 

On  the  other  hand  if  the  two  parallels  AA  and  BB' 
are  drawn  from  the  ends  of  the  segment  AB^  making  equal 
angles  with  AB^  we  have 


I  Using  the  result  tan 


TT  {x)  ^    --  xlk 


The  Square  of  area  it/t*. 


109 


<^  A  AB  ==  <:  B'BA  =  n  (^), 

where  AB  ==  r  [Fig.  52]. 

Now  draw  ^C,  perpendicular  to  BB\  and  ^Z>  perpen- 
dicular to  AC;  also  put 

<^  CAB  =  a,  <^  Z)^^'  =  z. 
Then  we  have 

cot  TT  (  ~  j  cot  a  +  I 


tan  z  =  cot  f  TT  r  '  j  —  cc  j 


cot  a  —  cot 


Ki) 


It  is  easy  to  eliminate  a  from  this  last  result  by  means 
of  the  trigonometrical  formulae  for  the  triangle  ABC  and  so 

obtain 

2 
tan  z  = — ^ 

.a„  n  (^) 

Substituting  this  in  the  expression  found  for  the  area  of 
the  circle,  we  obtain  for  that  area 

IT  k^  tan^  z. 
This  formula,  proved  in  an-  D 
other  way  by  Bolyai  {Appendix 
%  43],  allows  us  to  associate  a 
definite  angle  z  with  every  circle. 
If  3  were  equal  to  45",  then  we 
would  have 

for  the  area  of  the  correspond- 
ing circle. 


Fig.  52. 


2    C0t2 


I  Indeed,  in  the  rightangled  triangle  ABC,  we  have  cot  TT  (  —  ) 

=  cosh  _ 

k 

TT  (  "^  )  +  I'  we  deduce,  first,  that 


cosh  —    From  this,  since  cosh  -^   =  2  sinh2    L.   4-  i 

k  k  2k 


I  IO      IV.    The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

That  is  :  the  a7'ea  of  the  circle,  for  which  the  angle  z  is 
4j°,  is  equal  to  the  area  of  the  maximtim  triangle,  and  thus 
to  that  of  the  square  of  §  52. 

If  z  =  ^AAD  (Fig.  51)  is  given,  we  can  find  r  by 
the  following  construction: 

(i)     Draw  the  line  AC  perpendicular  to  AD. 

(ii)  Draw  BB'  parallel  to  AA  and  perpendicular  to 
^C7(S5i). 

(iii)  Draw  the  bisector  of  the  strip  between  AA  and 
BB\ 

[By  the  theorem  on  the  concurrency  of  the  bisectors  of 
the  angles  of  a  triangle  with  an  infinite  vertex.] 

(iv)  Draw  the  perpendicular  AB  to  this  bisector.  The 
segment  AB  bounded  by  AA  and  BB'  is  the  required 
radius  r. 

§  54.  The  problem  of  constructing  a  polygon  equal  to 
a  circle  of  area  tc  k'^  tan^  z  is,  as  Bolyai  remarked,  closely 
allied  with  the  numerical  value  of  tan  z.  It  is  resolvable 
for  every  integral  value  of  tan-  z,  and  for  every  fractional 
value,  provided  that  the  denominator  of  the  fraction,  re- 
duced to  its  lowest  terms,  is  included  in  the  form  assigned  by 
Gauss  for  the  construction  of  regular  polygons  [Appendix 

§43]- 

The  possibility  of  constructing  a  square  equal  to  a 
circle  leads  Johann  to  the  conclusion  "habeturque  aut  Axi- 
oma  XI  Euclidis  verutn,  aut  quadratura  circuii  geometrica; 

cot  TT  I  —  )  cot  a  =  2  cot2  TT  (  —  )  +1, 


and  next  that 
cot  a  —  cot 


TT(T)=(^+tan3n(;))cotn(^). 

These  equations  allow  the  expression  for  tan  z  to  be  written  down 
in  the  required  form. 


The  Quadrature  of  the  Circle.  Ill 

etsi  hucusque  indecisum  manserit,  quodnam  ex  his  duobus 
revera  locum  habeat." 

This  dilemma  seemed  to  him  at  that  time  [1831]  im- 
possible of  solution,  since  he  closed  his  work  with  these 
words:  "Superesset  denique  (ut  res  omni  numero  absolvatur), 
impossibilitatem  (absque  suppositione  aliqua)  decidenda, 
num  X  (the  Euclidean  system)  aut  aliquod  (et  quodam)  S  (the 
Non-Euclidean  system)  sit,  demonstrare  :  quod  tamen  occasi- 
oni magis  idoneae  reservatur." 

Johann,  however,  never  published  any  demonstration 
of  this  kind. 

§  55.  After  183 1  BoLYAi  continued  his  labours  at  his 
geometry,  and  in  particular  at  the  following  problems: 

1.  The  connection  between  Spherical  Trigonometry  and 
Non-Euclidean  Trigonometry. 

2.  Can  one  prove  rigorously  that  Euclid's  Axiom  is 
not  a  consequence  of  what  precedes  it  ? 

3.  The  volume  of  a  tetrahedron  in  Non-Euclidean  geo- 
metry. 

As  regards  the  first  of  these  problems,  beyond  estab- 
lishing the  analytical  relation  connecting  the  two  trigono- 
metries [cf.  LoBATSCHEWSKY,  p.  90],  BoLYAi  recognized  that 
in  the  Non-Euclidean  hypothesis  there  exist  three  classes  of 
Uniform  Surfaces^  on  which  the  Non-Euclidean  trigono- 
metry, the  ordinary  trigonometry,  and  spherical  trigonometry 
respectively  hold.  To  the  first  class  belong  planes  and  hyper- 
spheres  [surfaces  equidistant  from  a  plane];  to  the  second, 
the  paraspheres  [Lobatschewsky's  Horospheres]  ;  to  the 
third,  spheres.  The  paraspheres  are  the  limiting  case 
when  we  pass  from  the  hyperspherical  surfaces  to  the 
spherical.    This  passage  is  shown  analytically  by  making  a 


I  BoLYAl   seems  to  indicate  by  this  name  the  surfaces  which 
behave   as   planes,    with   respect   to  displacement  upon  themselves. 


I  [2      IV,     The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

certain  parameter,  which  appears  in  the  formulae,  vary  con- 
tinuously from  the  real  domain  to  the  purely  imaginary 
through  infinity  [cf  Taurinus,  p.  82]. 

As  to  the  second  problem,  that  regarding  the  impos- 
sibility of  demonstrating  Axiom  XT,  Bolyai  neither  succeeded 
in  solving  it,  nor  in  forming  any  definite  opinion  upon  it. 
For  some  time  he  believed  that  we  could  not,  in  any  way, 
decide  which  was  true,  the  Euclidean  hypothesis  or  the 
Non-Euclidean.  Like  Lobatschewsky,  he  relied  upon  the 
analytical  possibility  of  the  new  trigonometry.  Then  we  find 
Johann  returning  again  to  the  old  ideas,  and  attempting  a 
new  demonstration  of  Axiom  XI.  In  this  attempt  he  applies 
the  Non-Euclidean  formulae  to  a  system  of  five  coplanar 
points.  There  must  necessarily  be  some  relation  between 
the  distance  of  these  points.  Owing  to  a  mistake  in  his 
calculations  Johann  did  not  find  this  relation,  and  for  some 
time  he  believed  that  he  had  proved,  in  this  way,  the  false- 
hood of  the  Non-Euclidean  hypothesis  and  the  absolute  truth 
of  Axiom  XI .^ 

However  he  discovered  his  mistake  later,  but  he  did 
not  carry  out  further  investigations  in  this  direction,  as  the 
method,  when  applied  to  six  or  more  points,  would  have  in- 
volved too  comphcated  calculations. 

The  third  of  the  problems  mentioned  above,  that  re- 
garding the  tetrahedron,  is  of  a  purely  geometrical  nature. 
BoLYAi's  solutions  have  been  recently  discovered  and  pub- 


I  The  title  of  the  paper  which  contains  Johann's  demon- 
stration is  as  follows:  "Beweis  des  bis  mm  auf  der  Erde  itnmer 
nock  zwei/elkafi  gewesenen,  weltberuhmten  ujid,  ah  der  gesamtnten 
Raum-  und  Bewegungslehre  zu  Grunae  dienend,  auch  in  der  That 
allerh'òchst7uichtigsten  11.  Eudid'schen  Axioms  von  J.  Bolyai  von  Bolya, 
k.  k.Geiiie-Stabs/iauptmann  in  Pension.  Cf.  StaCKEL's  paper:  Untcr- 
suchungen  aiis  der  Absoluten  Geotnetrie  aus  yohann  Bolyais  N'achlafi. 
Math.  u.  Naturw.  Berichte  aus  Ungarn.  Ed.  XVIII,  p.  2S0— 307  (1902). 
We  are  indebted  to  this  paper  for  this  section  S  55- 


Bolyai^s  Later  Work.  1 1  5 

lished  by  Stackel  [cf.  p.  112  note  i].  Lobatschewsky 
had  been  often  occupied  with  the  same  problem  from  1829', 
and  Gauss  proposed  it  to  Johann  in  his  letter  quoted  on 
p.  100. 

Finally  we  add  that  J.  Bolyai  heard  of  Lobatschewsky's 
Geometrisc/ie  Untersuchimgen  in  1848:  that  he  made  them 
the  object  of  critical  study  ^:  and  that  he  set  himself  to  com- 
pose an  important  work  on  the  reform  of  the  Principles  of 
Mathematics  with  the  hope  of  prevailing  over  the  Russian. 
He  had  planned  this  work  at  the  time  of  the  publication  of 
the  Appendix,  but  he  never  succeeded  in  bringing  it  to  a 
conclusion.'' 

The  Absolute  Trigonometry. 

§  56.  Although  the  formulae  of  Non-Euclidean  trigono- 
metry contain  the  ordinary  relations  between  the  sides  and 
angles  of  a  triangle  as  a  limiting  case  [cf.  p.  80],  yet  they  do 
not  form  a  part  of  what  Johann  Bolyai  called  Absolute  Geo- 
metry. Indeed  the  formulse  do  not  apply  at  once  to  the  two 
classes  of  geometry,  and  they  were  deduced  on  the  suppos- 
ition of  the  validity  of  the  Hypothesis  of  the  Acute  Angle. 
Equations  directly  applicable  both  to  the  Euclidean  case  and 
to  the  Non-Euclidean  case  were  met  by  us  in  §  49  and  they 
make  up  Bolyai's  Theorem.  They  are  tliree  in  number,  only 
two  of  them  being  independent.  Thus  they  furnish  a  first 
set  of  formulae  of  Absolute  Trigonometry. 


»  Cf.  p.  53  et  seq.,  of  the  work  quoted  on  p.  84.  Also 
Liebmann's  translation,  referred  to  in  Note  2,  p.  85. 

2  Cf.  P.  Stackel  und  J.  KurschA'k:  Johann  Bolyals  Be- 
nierkungen  iiber  JV.  Lobaischefskijs  Geofneh-iscke  Untermchungen  zur 
Theorie  der  Parallellinicn,  Math.  u.  Naturw.  Berichte  aus  Ungarn, 
Bd.  XVIII,  p.  250—279  (1902). 

3  Cf.  P.  StAckel:  Johann  Bolyais  Raumlehre,  Math.  u.  Naturw. 
Berichte  aus  Ungarn,  Bd.  XIX  (1903). 

8 


jj_^      IV.    The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

Other  formulae  of  Absolute  Trigonometry  were  given  in 
1870  by  the  Belgian  geometer,  De  Tilly,  in  ins  Etudes  de 
Mécaniqiie  Abstraite.  ^ 

The  formulae  given  by  De  Tilly  refer  to  rectilinear  tri- 
angles, and  were  deduced  by  means  of  kinematical  con- 
siderationS;  requiring  only  those  properties  of  a  bounded 
region  of  a  plane  area,  which  are  independent  of  the  value 
of  the  sum  of  the  angles  of  a  triangle. 

In  addition  to  the  function  0-'*^>  which  we  have  already 
met  in  Bolyai's  formulae,  -those  of  De  Tilly  contain  another 
function  Ex  defined  in  the  following  way: 

Let  r  be  a  straight  line  and  /  the  equidistant  curve, 
distant  x  from  r.  Since  the  arcs  of  /  are  proportional  to  their 
projections  on  r,  it  is  clear  that  the  ratio  between  a  (recti- 
fied) arc  of  /  and  its  projection  does  not  depend  on  the 
length  of  the  arc,  but  only  on  its  distance  x  from  r.  De 
Tilly's  function  Ex  is  the  function  which  expresses  this  ratio. 

On  this  understanding,  the  Formulae  of  Absolute  Trigon- 
ometry for  the  right-angled  triangle  ABC  2^0.  as  follows: 
(i)  \C)a  =  Qjc  sin  A 

[0'^  =  O^sin^' 

(2)  fcos  A  =  Ea.  sin  B 
[cos  B  =  Eb.  sin  A 

(3)  Ec  =  Ea.  Eb. 
The  set  (i)  is  equivalent  to  Bolyai's 

Theorem  for  the  Right- Angled  Triangle. 
All  the  formulae  of  Absolute  Trigono- 
metry could  be  derived  by  suitable  com- 
bination of  these  three  sets.  In  particular,  for  the  right-angled 
triangle,  we  obtain  the  following  equation:  — 

I  Mémoires  couronnés  et  autres  Mémoires,  Acad,  royale  de 
Belgique.  T.  XXI  (1870).  Cf.  also  the  work  by  the  same  author: 
Essai  sur  les  principes  Jmtdamentaux  dc  la  p-rométrie  et  de  la  Mccanique, 
Mém.  de  la  Soc.  des  Se.  de  Bordeaux.     T.  Ili  (cah.  I)  (1878). 


The  Absolute  Trigonometry.  II  c 

O^a  {Ea  +  Eb.  Ec)  +  Q'-^-  ^Eb  +  Ec.  Ea) 
=  O'^  (^^  +  Ea.  Eb). 
This  can  be  regarded  as  equivalent  to  the  Theorem  of 
Pythagoras  in  the  Absolute  Geometry.^ 

§  57.  Let  us  now  see  how  we  can  deduce  the  results 
of  the  Euclidean  and  Non-Euclidean  geometries  from  the 
equations  of  the  preceding  article. 

Euclidean  Case. 

The  Equidistant  Curve  (/)  is  a  straight  line  [that  is,  Ex 
=  1],  and  the  perimeters  of  circles  are  proportional  to 
their  radii. 

Thus  the  equations  (i)  become 
(i')  {a  =  c  sin  A 

\b  =  c  sin  B. 

The  equations  (2)  give 
(2')  cos  A  =  sin  B,  cos  B  =  sin  A. 

Therefore  A  A^  B  =  90°. 

Finally  the  equation  (3)  reduces  to  an  identity. 

The  equations  (i')  and  (2')  include  the  whole  of  ordin- 
ary trigonometry. 

Non- Euclidean  Case. 

Combining  the  equations  (i)  and  (2)  we  obtain 

E^a—l  E2b—\ 

If  we  now  apply  the  first  of  equations  (2)  to  a  right- 
angled  triangle  whose  vertex  A  goes  oft"  to  infinity,  so  that 
the  angle  A  tends  to  zero,  we  shall  have 

Lt  cos  A  =  Lt  {Ea.  sin  B). 
But  Ea  is  independent  of  A;  also  the  angle  B,  in  the 
limit,  becomes  the  angle  of  parallelism  corresponding  to  a, 
i.  e.  n  {a). 


^   Cf.    R.  BoNOLV,    La    trigonometria    assoluta    secondo    Giovann: 
Bolyai.     Rend.  Istituto  Lombardo  (2).     T.  XXXVIII  (1905). 

8* 


Il6      IV.    The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

Therefore  we  have 

sin  n  (a) 
A  similar  result  holds  for  Eb. 

Substituting  these  in  equation  (5)  we  obtain 

cot2  TT  {a)  cot2  TT  {b)  ' 

from  which 

cot  IT  {a)  cot  jr  (^} 

This  result,  with  the  expression  for  Ex,  allows  us  at 
once  to  obtain  from  the  equations  (i),  (2),  (3),  the  formulae 
of  the  Trigonometry  of  Bolyai-Lobatschewsky: 

fcot  TT  {a)  =  cot  TT  {c)  sin  A 
^^  '  jcot  n  {b)  =  cot  n  {c)  sin  B, 
,  „^  fsin  A  =  cos  B  sin  TT  {b) 

I  2     )  -I 

Ì  sin  j9  =  cos  A  sin  TT  («;), 
(3")  sin  TT  {c)  =  sin  TT  {a)  sin  TT  {b). 

These  relations  bet\veen  the  elements  of  every  right- 
angled  triangle  were  given  in  this  form  by  Lobatschewsky/ 
If  we  wish  to  introduce  direct  functions  of  the  sides,  instead 
of  the  angles  of  parallelism  TT  (a),  TT  {b)  and  TT  (^),  it  is 
sufficient  to  remember  [p.  90]  that 

tan  —^  =  e  "'*. 

We  can  thus  express  the  circular  functions  of  TT  {x)  in 
terms  of  the  hyperbolic  functions  of  x.  In  this  way  the  pre- 
ceding equations  are  replaced  by  the  following  relations: 

(i"'_)  sinh  -7-  =  sinh  -j  sin  A 


k         k 

b 
J 


sinh  -r  =  sinh  -7-  sin  B, 


I  Cf.  e.  g.,  The  Geometrische  Untersuchungen  of  LOBATSCHEWSKY 
referred  to  on  p.  86. 


Absolute  Trigonometry  and  Spherical  Trigonometry.       117 

(2"')  COS  A  =  sin  B  cosh  -r 

cos  B  =  sin  A  cosh  ^r. 


and 

(•?'")  cosh  -,-   =   cosh  -y-  cos  /i  cosh  -7-. 

§  58.  The  following  remark  upon  Absolute  Trigono- 
metry is  most  important:  //  we  regard  the  elements  in  its 
formulae  as  elements  of  a  spherical  triangle,  we  obtain  a  system 
of  equations  which  hold  also  for  Spherical  Triangles. 

This  property  of  Absolute  Trigonometry  is  due  to  the 
fact,  already  noticed  on  p.  114,  that  it  was  obtained  by  the 
aid  of  equations  which  hold  only  for  a  limited  region  of  the 
plane.  Further  these  do  not  depend  on  the  hypothesis  of  the 
angles  of  a  triangle,  so  that  they  are  valid  also  on  the  sphere. 

If  it  is  desired  to  obtain  the  result  directly,  it  is  only 
necessary  to  note  the  following  facts: — 

(i)  In  Spherical  Trigonometry  the  circumferences  of 
circles  are  proportional  to  the  sines  of  their  (spherical)  radii, 
so  that  the  first  formula  for  right-angled  spherical  triangles 

sin  <?  =  sin  ^  sin  A 
is  transformed  at  once  into  the  first  of  the  equations  (i), 

(ii)  A  circle  of  (spherical)  radius b  can  be  [con- 
sidered as  a  curve  equidistant  from  the  concentric  great 
circle,  and  the  ratio  Eb  for  these  two  circles  is  given  by 


(v  -  0 


sm  J^ 
2 


=  cos  b. 


Thus  the  formulae  for  right-angled  spherical  triangles 
cos  ^  =  sin  ^  cos  a, 
cos  c  =  cos  a  cos  b. 


jl8    IV.     The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

are  transformed  immediately  into  the  equations  (2)  and  (3) 
by  means  of  this  result. 

Thus  the  formulae  of  Absolute  Trigonometry  also  hold 
on  the  sphere. 

Hypotheses  equivalent  to  Euclid's  Postulate. 

§  59.  Before  leaving  the  elementary  part  of  the  sub- 
ject, it  seems  right  to  call  the  attention  of  the  reader  to  the 
position  occupied  in  the  general  system  of  geometry  by  certain 
propositions,  which  are  in  a  certain  sense  hypotheses  equivalent 
to  the  Fifth  Postulate. 

That  our  argument  may  be  properly  understood,  we 
begin  by  explaining  the  meaning  of  this  equivalence. 

Two  hypotheses  are  absolutely  equivaletit  when  each  of 
them  can  be  deduced  from  the  other  without  the  lielp  of  any 
new  hypothesis.  In  this  sense  the  two  following  hypotheses 
are  absolutely  equivalent: 

a)  Two  straight  lines  parallel  to  a  third  are  parallel  to 
each  other; 

b)  Through  a  point  outside  a  straight  line  one  and  only 
one  parallel  to  it  can  be  drawn. 

This  kind  of  equivalence  has  not  much  interest,  since 
the  two  hypotheses  are  simply  two  different  forms  of  the 
same  proposition.  However  we  must  consider  in  what  way 
the  idea  of  equivalence  can  be  generalised. 

Let  us  suppose  that  a  deductive  science  is  founded 
upon  a  certain  set  of  hypotheses,  which  we  will  denote  by 
\A,B,  C,...If\.  'LttM  and  ^be  two  new  hypotheses  such 
that  N  can  be  deduced  from  the  set  [A,  B,  C ...  IT,  J/|, 
and  M  from  the  set  {a,  B,  C .  .  .  H,  N) 

We  indicate  this  by  writing 

{A,B,  C ...H,M)  .)  .jV, 


Absolute  and  Relative  Equivalence.  ng 

and 

{A,B,  C ..  .H,N)  .).  M. 

We  shall  now  extend  the  idea  of  equivalence  and  say  that 
the  two  hypotheses  J/,  N  are  equivalent  relatively  to  the 
fundamental  set  \A,  B,  C .  .  .  HY 

It  has  to  be  noted  that  the  fundamental  set  {A,  B,  C 
.  .  .  Jl^f  has  an  important  place  in  this  definition.  Indeed  it 
might  happen  that  by  diminishing  this  fundamental  set,  leav- 
ing aside,  for  example,  the  hypothesis  A,  the  two  deductions 

{B,  C,.,.JI,M}  .).  JV 
and 

{B,C,...If,Ar\.).M 

could  not  hold  simultaneously. 

In  this  case  the  hypotheses  M,  N  are  not  equivalent 
with  respect  to  the  new  fundamental  set  \B^  C  .  .  .  H^ 

After  these  explanations  of  a  logical  kind,  let  us  see 
what  follows  from  the  discussion  in  the  preceding  chapters 
as  to  the  equivalence  between  such  hypotheses  and  the 
Euclidean  hypothesis. 

We  assume  in  the  first  place  as  fundamental  set  of 
hypotheses  that  formed  by  the  postulates  of  Association  {A), 
and  of  Distribution  {B)^  which  characterise  in  the  ordinary 
way  the  conceptions  of  the  straight  line  and  the  plane:  also 
by  the  postulates  of  Congruence  (C),  and  the  Postulate  of 
Archimedes  (Z>). 

Relative  to  this  fundamental  set,  which  we  shall  denote 
by  \a,  B,  C,  D\,  the  following  hypotheses  are  mutually 
equivalent,  and  equivalent  also  to  that  stated  by  Euclid  in 
his  Fifth  Postulate: 

a)  The  internal  angles,  which  two  parallels  make  with  a 
transversal  on  the  same  side,  are  supplementary  [Ptolemy]. 

b)  Two  parallel  straight  lines  are  equidistant. 

c)  If  a  straight  line  intersects  one  of  two  parallels,  it 
also  intersects  the  other  (Proclus); 


120      IV.    The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

or, 

Two  straight  lines,  which  are  parallel  to  a  third,  are 
parallel  to  each  other; 
or  again. 

Through  a  point  outside  a  straight  line  there  can  be 
drawn  one  and  only  one  parallel  to  that  line. 

d)  A  triangle  being  given,  another  triangle  can  be  con- 
structed similar  to  the  given  one  and  of  any  size  whatever. 
[Wallis.] 

e)  Through  three  points,  not  lying  on  a  straight  line,  a 
sphere  can  always  be  drawn.    [W.  Bolyai.] 

f)  Through  a  point  between  the  Hnes  bounding  an  angle 
a  straight  hne  can  always  be  drawn  which  will  intersect  these 
two  lines.    [Lorenz.] 

a)  If  the  straight  line  r  is  perpendicular  to  the  trans- 
versal AB  and  the  straight  line  s  cuts  it  at  an  acute  angle? 
the  perpendiculars  from  the  points  of  s  upon  r  are  less  than 
AB^  on  the  side  in  which  AB  makes  an  acute  angle  with  s. 
[Nasìr-Eddìn.] 

P)  The  locus  of  the  points  which  are  equidistant  from 
a  straight  line  is  a  straight  line. 

f  )  The  sum  of  the  angles  of  a  triangle  is  equal  to  two 
right  angles.    [Saccherl] 

Now  let  us  suppose  that  we  diminish  the  fundamental 
set  of  hypotheses,  cutii?ig  oiit  the  Archimedean  Hypothesis. 
Then  the  propositions  (a),  (b),  (c),  (d),  (e)  and  (f)  are 
mutually  equivalent,  and  also  equivalent  to  the  Fifth  Postu- 
late of  Euclid,  with  respect  to  the  fundamental  set  |^,  ^,  C]. 
With  regard  to  the  propositions  (a),  (P),  (t),  while  they  are 
mutually  equivalent  with  respect  to  the  set  \A,  B,  C|  no  one 
of  them  is  equivalent  to  the  Euclidean  Postulate.  This  result 
brings  out  clearly  the  importance  of  the  Postulate  of  Archi- 
medes.  It  is  given  in  the  memoir  of  Dehn'  [19°°]  to  which 

I  Cf.  Note  on  p.  30. 


Hypotheses  Equivalent  to  Euclid's  Postulate.  i2I 

reference  has  already  been  made.  In  that  memoir  it  is  sho^vn 
that  the  hypothesis  (f)  on  the  sum  of  the  angles  of  a  triangle 
is  compatible  not  only  with  the  ordinary  elementary  geo- 
metry, but  also  with  a  new  geometry— necessarily  Non-Archi- 
medean—where the  Fifth  Postulate  does  not  hold,  and  in 
which  an  infinite  number  of  lines  pass  through  a  point  and 
do  not  intersect  a  given  straight  line.  To  this  geometry  the 
author  gave  the  name  of  Semi-Euclidean  Geometry. 

The  Spread  of  Non-Euclidean  Geometry. 

§  60.  The  works  of  Lobatschewsky  and  Bolyai  did 
not  receive  on  their  publication  the  welcome  which  so  many 
centuries  of  slow  and  continual  preparation  seemed  to 
promise.  However  this  ought  not  to  surprise  us.  The 
history  of  scientific  discovery  teaches  that  every  radical  change 
in  its  separate  departments  does  not  suddenly  alter  the  con- 
victions and  the  presuppositions  upon  which  investigators 
and  teachers  have  for  a  considerable  time  based  the  present- 
ation of  their  subjects. 

In  our  case  the  acceptance  of  the  Non-Euclidean  Geo- 
metry was  delayed  by  special  reasons,  such  as  the  difficulty 
of  mastering  Lobatschewsky's  works,  written  as  they  were  in 
Russian,  the  fact  that  the  names  of  the  two  discoverers  were 
new  to  the  scientific  world,  and  the  Kantian  conception  of 
space  which  was  then  in  the  ascendant. 

Lobatschewsky's  French  and  German  writings  helped 
to  drive  away  the  darkness  in  which  the  new  theories  were 
hidden  in  the  first  years;  more  than  all  availed  the  constant 
and  indefatigable  labors  of  certain  geometers,  whose  names 
are  now  associated  with  the  spread  and  triumph  of  Non- 
Euclidean  Geometry.  We  would  mention  particularly:  C.  L. 
Gerling  [1788— 1864],  R.  Baltzer  [1818— 1887]  and  Fr. 
Schmidt   [1827  — 1901],  in  Germany;   J.   Hoùel  [1823 — 


122    IV.     The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

1886],  G.  Battaglini  [1826 — 1894],  E.  Beltrami  [1835— 
1900],  and  A.  Forti,  in  France  and  Italy. 

§  61.  From  181 6  Gerling  kept  up  a  correspondence 
upon  parallels  with  Gauss  %  and  in  181 9  he  sent  him 
Schweikart's  memorandum  on  Astra lgeo?/ietrie  [cf.  p.  75]. 
Also  he  had  heard  from  Gauss  himself  [1832],  and  in  terms 
which  could  not  help  exciting  his  natural  curiosity,  of  a 
kleine  Schrift  on  Non-Euclidean  Geometry  written  by  a 
young  Austrian  officer,  son  of  W.  Bolyai.*  The  bibliograph- 
ical notes  he  received  later  from  Gauss  [1844]  on  the  works 
of  Lobatschewsky  andBoLYAi^  induced  Gerling  to  procure 
for  himself  the  Geomdrischeii  Uiitersuchtingen  and  the  Appen- 
dix, and  thus  to  rescue  them  from  the  oblivion  into  which 
they  seemed  plunged. 

§  62.  The  correspondence  between  Gauss  and  Schu- 
macher, published  between  i860  and  iSós,"^  the  numerous 
references  to  the  works  of  LoBATSCHEWSKy  and  Bolyai,  and 
the  attempts  of  Legendre  to  introduce  even  into  the  elemen- 
tary text  books  a  rigorous  treatment  of  the  theory  of  pa- 
rallels, led  Baltzer,  in  the  second  edition  of  his  Elemmte  der 

1  Cf.  Gauss,  Werke,  Bd.  VIII,  p.  167—169. 

2  Cf.  Gauss's  letter  to  Gerling  (Gauss,  Werke,  Bd.  VIII, 
p.  220).  In  this  note  Gauss  says  with  reference  to  the  contents 
of  the  Appendix:  "worin  ich  alle  meine  eigenen  Ideen  taid  Resultate 
wlederfiiìde  mit  g7-ofier  Eleganz  entwickelt."  And  of  the  author  of 
the  work  :  „Ich  halte  diesm  jiingC7i  Geonietei'  v.  Bolyai  fib-  eni  Genie 
erster  Grafie". 

3  Cf.  Gauss,  IVerke,  Bd.  VIII,  p.  234—238. 

4  Briefwechsel  ziuischen  C,  F.  Gauss  7cnd  H.  C.  Schuinacher, 
Bd.  II,  p.  268—431  Bd.  V,  p.  246  (Altona,  1860—1863).  As  to 
Gauss's  opinions  at  this  time,  see  also,  Sartorius  von  Walters' 
HaUSEN,  Gatifi  zutn  Geddr/Unis,  p.  8o— 8l  (Leipzig,  1S56).  Cf.  GAUSS, 
Werke,  Bd.  VIII,  p.  267—268. 


The  Spread  of  Non-Euclidean  Geometry.  1 23 

Mathemafik  {1^6'j),  to  substitute,  for  the  Euclidean  definition 
of  parallels  one  derived  from  the  new  conception  of  space. 
Following  LoBATSCHEWSKY  he  placed  the  equation  A-\-B 
+  C  =  180°,  which  characterises  the  Euclidean  triangle, 
among  the  experimental  results.  To  justify  this  innovation, 
Baltzer  did  not  fail  to  insert  a  brief  reference  to  the  possi- 
bility of  a  more  general  geometry  than  the  ordinary  one, 
founded  on  the  hypothesis  of  two  parallels.  He  also  gave 
suitable  prominence  to  the  names  of  its  founders.^  At  the 
same  time  he  called  the  attention  of  Houel,  whose  interest 
in  the  question  of  elementary  geometry  was  well  known  to 
scientific  men,  ^  to  the  Non-Euclidean  geometry,  and  re- 
quested him  to  translate  the  Geometrischen  Untersiichungen 
and  the  Appendix  into  French. 

§  63.  The  French  translation  of  this  little  book  by 
LoBATSCHEWSKY  appeared  in  1866  and  was  accompanied 
by  some  extracts  from  the  correspondence  between  Gauss 
and  Schumacher.^  That  the  views  of  Lobatschewsky, 
Bolyai,  and  Gauss  were  thus  brought  together  was  extremely 
fortunate,  since  the  name  of  Gauss  and  his  approval  of  the 
discoveries  of  the  two  geometers,  then  obscure  and  unknown, 

1  Cf.  Baltzer,  Elemente  der  Mathematik,  Bd.  11  (5.  Auflage) 
p.  12 — 14  (Leipzig,  1878).  Also  T.  4,  p.  5 — 7,  of  Cremona's  trans- 
lation of  that  work  (Genoa,   1867). 

2  In  1863  HoiJEL  had  published  his  wellknown  Essai  d'une 
exposition  7-ationelIe  des  principes  fondametitmcx  de  la  Geometrie  èli- 
7!ientaire.     Archiv  d.  Math.  u.  Physik,  Bd.  XL  (1863). 

3  Ména,  de  la  Soc.  des  Sci.  de  Bordeaux,  T.  IV,  p.  88 — 120 
(1S66).  This  short  work  was  also  published  separately  under  the 
title  Etudes  géométriques  sur  la  théorie  des  parallèles  par  N.  I.  LoBAT- 
SCHEWSKY, Conseiller  d'État  de  l'Empire  de  Russie  et  Professeur 
à  rUniversité  de  Kasan:  traduit  de  l'allemand  par  J.  Houel,  suivie 
d'un  Extrait  de  la  correspondance  de  Gauss  et  de  Schumacher,  (Paris, 
G.   VU.LARS,    1866). 


124      ^^'  T^ti^  Founders  of  Non-Euclidean  Geometry  (Contd.). 

helped  to  bring  credit  and  consideration  to  the  new  doctrines 
in  the  most  efficacious  and  certain  manner. 

The  French  translation  of  the  Appendix  appeared  in 
1867.'  It  was  preceded  by  a  Notice  sur  la  vie  et  les  travaux 
des  deux  viathématiciens  hotigrois  W.  et  J.  Bolyai  de  Bolya, 
written  by  the  architect  Fr.  Schmidt  at  the  invitation  of 
HoiJEL,^  and  was  supplemented  by  some  remarks  by  W.  Bol- 
yai, taken  from  Vol.  I  of  the  Tentameli  and  from  a  short 
analysis,  also  by  Wolfgang,  of  the  Principles  of  Arithmetic 
and  Geometry.3 

In  the  same  year  [1867]  Schmidt's  discoveries  regard- 
ing the  BoLYAis  were  published  in  the  Archiv  d.  Math.  u. 
Phys.  Also  in  the  following  year  A.  Forti,  who  had  already 
written  a  critical  and  historical  memoir  on  Lobatschewsky,'* 


1  Mém.  de  la  Soc.  des  Se.  de  Bordeaux,  T.  V,  p.  189 — 
248.  This  short  work  was  also  published  separately  unter  the 
title:    La  Science  absolute    de    l' espace,    indèpendante    de    la    vérité    on 

fausseti  de  l'Axiome  XI  d'Euclide  {que  l'on  ne  pourra  jamais  établiì-  a 
priori);  suivie  de  la  quadrature  géometrique  du  cercle,  dans  le  cas  de 
la  fausseté  de  l'Axiome  XI,  par  Jean  Bolyai,  Capitaine  au  Corps 
du  genie  dans  l'armée  autrichienne;  Précède  d'iene  notice  sur  la  vie 
it  les  travaux  de  W.  et  de  J.  Bolyai,  par  M.  Fr.  Schmidt,  (Paris, 
G.   VlLLARS,    1868). 

2  Cf.  P.  StAckel,  Franz  Schmidt,  Jahresber.  d.  Deutschen 
Math.  Ver.,  Bd.  XI,  p.  141  —  146  (1902). 

3  This  little  book  of  \V.  BoLYAl's  is  usually  referred  to 
shortly  by  the  first  words  of  the  title  Kicrzer  Grtmdriss.  It  was  pub- 
lished at  Maros-Visàrhely  in  1851. 

■*  Intorno  alla  geometria  itnmaginaria  o  non  euclidiana.  Consid- 
erazioni storico-critiche.  Rivista  Bolognese  di  scienze,  lettere,  arti 
e  scuole,  T.  Il,  p.  171  — 184  (1867).  It  was  published  separately 
as  a  pamphlet  of  16  pages  (Bologna,  Fava  e  Garagnani,  1867). 
The  same  article,  with  some  additions  and  the  title,  Studii  geo- 
metrici sulla  teorica  delle  parallele  di  N.  J.  Lobatschewky,  appeared 
in  the  politicai  journal  La  Provincia  di  Pisa,  Anno  III,   Nr.  25,  27, 


Hoùel  and  Schmidt.  1 25 

made  the  name  and  the  works  of  the  two  now  celebrated 
Hungarian  geometers  known  to  the  Italians/ 

To  the  credit  of  Hoùel  there  should  also  be  mentioned 
his  interest  in  the  manuscripts  of  Johann  Bolyai,  then  [1867] 
preserved,  in  terms  of  Wolfgang's  will,  in  the  library  of 
the  Reformed  College  of  Maros-Vàsàrhely.  By  the  help  of 
Prince  B.  Boncampagni  [182  i — 1894],  who  in  his  turn  in- 
terested the  Hungarian  Minister  of  Education,  Baron  Eòtvòs, 
he  succeeded  in  having  them  placed  [1869]  in  the  Hungarian 
Academy  of  Science  at  Budapest.^  In  this  way  they  became 
more  accessible  and  were  the  subject  of  painstaking  and 
careful  research,  first  by  Schmidt  and  recently  by  Stackel. 

In  addition  Houel  did  not  fail  in  his  efforts,  on  every 
available  opportunity,  to  secure  a  lasting  triumph  for  the  Non- 
Euclidean  Geometry.  If  we  simply  mention  his  Essai  cri- 
tique sur  les principes  fondameìiteaux  de  la  geometrie:'^  his  ar- 
ticle, Sur  l' impossibilité  de  démontrer  par  tene  construction 
plane  le  postulatum  d'Euclide;  "*  the  Notices  sur  la  vie  et  les 
iravaux  de  N.  J.  Lobatschewsky;  5  and  finally  his  translations 
of  various  writings  upon  Non-Euclidean  Geometry  into  French,^ 


29,  30  (1867);  and  part  of  it  was  reprinted  under  the  original  title 
(Pisa,  Nistri,   1867). 

*  Cf.  Iniorito  alia  vita  ed  agli  sa-itti  di  Wolfgang  e  Giovanni 
Bolyai  di  Bolya,  rnatemalici  ungheresi.  Boll,  di  Bibliografia  e  di 
Storia  delle  Scienze  Mat.  e  Fisiche.  T.  I,  p.  277—299  (1869), 
Many  historical  and  bibliographical  notes  were  added  to  this  article 
of  Forti's  by  B.  Boncompagni. 

2  Cf.  Stackel's  article  on  Franz  Schmidt  referred   to    above. 

3  I.  Ed.,  G.  ViLLARS,  Paris,  1867;  2  Ed.,  1883  (cf.  Note  3 
p.  52). 

4  Giornale  di  Mat.  T.  VII  p.  84— 89;  Nouvelles  Annales  (2) 
T.  IX,  p.  93-96. 

5  Bull.  des.  Sc.  Math.  T.  I,  p.  66—71,  324—328,  384—388 
(1870). 

0  In  addition  to  the  translations  mentioned  in  the  text,  Hoùel 


126   IV.  The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

it  will    e  understood  how  fervent  an  apostle  this  science  had 
found  in  the  famous  French  mathematician. 

Hoùel's  labours  must  have  urged  J.  Frischauf  to  per- 
form the  service  for  Germany  which  the  former  had  rendered 
to  France.  His  book — Absolute  Geometrie  nach  J.  Bolyai  — 
(1872)"  is  simply  a  free  translation  of  Johann's  Appendix,  to 
which  were  added  the  opinions  of  W.  Bolyai  on  the  Found- 
ations of  Geometry.  A  new  and  revised  edition  of  Frisch- 
auf's  work  was  brought  out  in  1876^.  In  that  work  reference 
is  made  to  the  writings  of  Lobatschewsky  and  the  memoirs 
of  other  authors  who  about  that  time  had  taken  up  this  study 
from  a  more  advanced  point  of  view.  This  volume  remained 
for  many  years  the  only  book  in  which  these  new  doctrines 
upon  space  were  brought  together  and  compared. 

§  64.  With  equal  conviction  and  earnestness  Giuseppe 
Battaglini  introduced  the  new  geometrical  speculations  into 
Italy  and  there  spread  them  abroad.  From  1867  the  Gior- 
nale di  Matematica,  of  which  he  was  both  founder  and  editor, 
became  the  recognized  organ  of  Non-Euclidean  Geometry. 

Battaglini's  first  memoir — Sulla  geometria  immaginaria 
di  Lobatschewsky^— y^z.%  written  to  establish  directly  the  prin- 
ciple which  forms  the  foundation  of  the  general  theory  of 
parallels  and  the  trigonometry  of  Lobatschewsky.    It  was 


translated  a  paper  by  Battaglini  (cf.  note  3),  two  by  Beltrami 
(cf.  note  2  p.  127  and  p.  147);  one,  by  Rif.mann  (cf.  note  p.  138). 
and  one  by  Helmholtz  (cf.  note  p.   152). 

1  (xii  ■\-  96  pages)  (Teubner,  Leipzig). 

2  Eletnente  der  Ahsoluteii  Geometrie,  (vi  -|-  142  pages)  (Teubner, 
Leipzig). 

3  Giornale  di  Mat.  T.  V,  p.  217 — 231  (1S67).  Rend.  Ace. 
Science  Fis.  e  Matem.  Napoli,  T.  VI,  p.  157 — 173  (1867).  French 
translation,  by  HoUEL,  Nouvelles  Annales  (2)  T.  VII,  p.  209—21, 
2Ó5— 277  (i8óS). 


Battaglini  and  Beltrami.  127 

followed,  a  few  pages  later,  by  the  Italian  translation  of  the 
Pangéométrie'^;  and  this,  in  its  turn,  in  1868,  by  the  translation 
of  the  Appendix. 

At  the  same  time,  in  the  sixth  volume  of  the  Giornale  di 
Matematica,  appeared  E.  Beltrami's  famous  paper,  Saggio  di 
ititerpretazione  della  geometria  non  euclidea.  ^  This  threw  an 
unexpected  light  on  the  question  then  being  debated  regard- 
ing the  fundamental  principles  of  geometry,  and  the  concep- 
tions of  Gauss  and  Lobatschewsky.-^ 

Glancing  through  the  subsequent  volumes  of  the  Giorn- 
ale di  Matematica  we  frequently  come  upon  papers  upon 
Non-Euclidean  Geometry.  There  are  two  by  Beltrami  [1872] 
connected  with  the  above— named  Saggio;  several  by  Batt- 
aglini [1874 — 78]  and  by  d'OviDio  [1875 — 77]?  which  treat 
some  questions  in  the  new  geometry  by  the  projective  me- 
thods discovered  by  Cayley;  Houel's  paper  [1870]  on  the 
impossibility  of  demonstrating  Euclid's  Postulate;  and  others 
by  Cassani  [1873 — 81],  Gunther  [1876],  De  Zolt  [1877], 
Frattini  [1878],  Ricordi  [1880],  etc. 

§  65.  The  work  of  spreading  abroad  the  knowledge  of 
the  new  geometry,  begun  and  energetically  carried  forward 
by  the  aforesaid  geometers,  received  a  powerful  impulse  from 
another  set  of  publications,  which  appeared  about  this  time 
[1868—72].  These  regarded  the  problem  of  the  foundations 
of  geometry  in  a  more  general  and  less  elementary  way  than 
that  which  had  been  adopted  in  the  investigations  of  Gauss, 


1  This  was  also  published  separately  as  a  small  book,  entitled, 
Pangeometria  0  sunto  di  geometria  fondata  sopra  una  teoria  generate 
e  rigorosa  delle  parallele  (Naples,    1867;  2a  Ed.   1874). 

2  It  was  translated  into  French  by  Houel  in  the  Ann.  Sc.  de 
l'École  Normale  Sup.,  T.  VI,  p.  251—288  (1869). 

3  Cf.  Commemorazione  di  E.  Beltrami  by  L.  CREMONA:  Giornale 
di  Mat.  T.  XXXVIII,  p.  362  (1900).  Also  the  Nachruf  by  E. 
Pascal,  Math.  Ann.  Bd.  LVII,  p.  65—107  (1903). 


128    IV.  The  Founders  of  Non-Euclidean  Geometry  (Contd.). 

LoBATSCHEWSKY,  and  BoLYAi.  In  Chapter  V.  we  shall  shortly 
describe  these  new  methods  and  developments,  which  are  asso- 
ciated with  the  names  of  some  of  the  most  eminent  mathe- 
maticians and  philosophers  of  the  present  time.  Here  it  is 
sufficient  to  remark  that  the  old  question  of  parallels,  from 
which  all  interest  seemed  to  have  been  taken  by  the  in- 
vestigations of  Legendre  forty  years  earlier,  once  again  and 
under  a  completely  new  aspect  attracted  the  attention  of  geo- 
meters and  philosophers,  and  became  the  centre  of  an 
extremely  wide  field  of  labour.  Some  of  these  efforts  were 
simply  directed  toward  rendering  the  works  of  the  founders 
of  Non-Euclidean  geometry  more  accessible  to  the  general 
mathematical  public.  Others  were  prompted  by  the  hope  of 
extending  the  results,  the  content,  and  the  meaning  of  the 
new  doctrines,  and  at  the  same  time  contributing  to  the  pro- 
gress of  certain  special  branches  of  Higher  Mathematics,^ 


I   Cf.   e.   g.,    É.   Picard,    La    Science    Moderne    et    son    état 
actual,  p.  75  (Paris,  Flammarion,  1905). 


Chapter  V. 

The  Later  Development  of  Non-Euclidean 
Geometry. 

§  66.  To  describe  the  further  progress  of  Non-Eudidean 
Geometry  in  the  direction  of  Differential  Geometry  and  Pro- 
jective Geometry,  we  must  leave  the  field  of  Elementary  Mathe- 
matics and  speak  of  some  of  the  branches  of  Higher  Mathe- 
matics, such  as  the  Differential  Geometry  of  Manifolds,  the 
Theory  of  Continuous  Transformation  Groups^  Pure  Projec- 
tive Geometry  (the  system  of  Staudt)  and  the  Metrical 
Geometries  which  are  subordinate  to  it.  As  it  is  not  consistent 
with  the  plan  of  this  work  to  refer,  even  shortly,  to  these 
more  advanced  questions,  we  shall  confine  ourselves  to  those 
matters  without  which  the  reader  could  not  understand  the 
motive  spirit  of  the  new  researches,  nor  be  led  to  that  other 
geometrical  system,  due  to  Riemann,  which  has  been  alto- 
gether excluded  from  the  previous  investigations,  as  they 
assume  that  the  straight  line  is  of  infinite  length. 

This  system  is  known  by  the  name  of  its  discoverer  and 
corresponds  to  the  Hypothesis  of  the  Obtuse  A?igle  of  Sac- 
CHERi  and  Lambert.^ 


^  The  reader,  who  wishes  a  complete  discussion  of  the  sub- 
ject of  this  chapter,  should  consult  Klein's  Vorlesungen  uber  die 
iiickteuklidische  Geometrie,  (Gòttingen,  1903);  and  BlANCHl's  Lezioni 
sulla  Geometria  differenziale,  2  Ed.  T.  I,  Cap.  XI  — XIV  (Pisa,  Spoerri, 
1903).  German  translation  by  Lukat,  i^t  Ed.  (Leipzig,  1899).  Also 
The  Elements  of  Non-Eicclidean  Geometry  by  T.  L.  CoOLlDGE  which 
has  recently  (1909)  been  published  by  the  Oxford  University  Press. 

9 


I  20    V.  The  Later  Development  of  Non-Euclidean  Geometry. 

Differential  Geometry  and  Non-Euclidean  Geometry. 

The  Geometry  upon  a  Surface. 

§  67.  What  follows  will  be  more  easily  understood  if 
we  start  with  a  few  observations: 

A  surface  being  given,  let  us  see  how  far  we  can  establish 
a  geometry  upon  it  analogous  to  that  on  the  plane. 

Through  two  points  A  and  B  on  the  surface  there  will 
generally  pass  one  definite  line  belonging  to  the  surface, 
namely,  the  shortest  distance  on  the  surface  between  the  two 
points.  This  line  is  called  the  geodesic  joining  the  two  points. 
In  the  case  of  the  sphere,  the  geodesic  joining  two  points,  not 
the  extremities  of  a  diameter,  is  an  arc  of  the  great  circle 
through  the  two  points. 

Now  if  we  wish  to  compare  the  geometry  upon  a  surface 
with  the  geometry  on  a  plane,  it  seems  natural  to  make  the 
geodesies,  which  measure  the  distances  on  the  one  surface, 
correspond  to  the  straight  lines  of  the  other.  It  is  also  natural 
to  consider  two  figures  traced  upon  the  surface  as  {geodetical- 
ly)  equal,  when  there  is  a  point  to  point  correspondence  be- 
tween them,  such  that  the  geodesic  distances  between  corre- 
sponding points  are  equal. 

We  obtain  a  representation  of  this  conception  of  equality, 
if  we  assume  that  the  surface  is  made  of  z.  flexible  and  itiex-, 
tensible  sheet.  Then  by  a  movement  of  the  surface,  which  does 
not  remain  rigid,  but  is  bent  as  described  above,  those  figures 
upon  it,  which  we  have  called  equal,  are  to  be  superposed 
the  one  upon  the  other. 

Let  us  take,  for  example,  a  piece  of  a  cylindrical  surface. 
By  simple  bending,  without  stretching,  folding,  or  tearing,  this 
can  be  applied  to  a  plane  area.  It  is  clear  that  in  this  case 
two  figures  ought  to  be  called  equal  on  the  surface,  which 
coincide  with  equal  areas  on  the  plane,  though  of  course  two 
such  figures  are  not  in  general  equal  in  space. 


Differential  Geometry  and  Non-Euclidean  Geometry.       j^j 

Returning  now  to  any  surface  whatsoever,  the  system  of 
conventions,  suggested  above,  leads  to  a  geometry  on  the  sur- 
face, which  we  propose  to  consider  ahvays  for  suitably  bounded 
regions  {^Normal Regions].  Two  surfaces  which  are  applicable 
the  one  to  the  other,  by  bending  without  stretching,  will  have 
the  same  geometry.  Thus,  for  example,  upon  any  cylindrical 
surface  whatsoever,  we  will  have  a  geometry  similar  to  that  on 
any  plane  surface,  and,  in  general,  upon  any  developable  surface. 

The  geometry  on  the  sphere  affords  an  example  of  a 
geometry  on  a  surface  essentially  different  from  that  on  the 
plane,  since  it  is  impossible  to  apply  a  portion  of  the  sphere 
to  the  plane.  However  there  is  an  important  analogy  be- 
tween the  geometry  on  the  plane  and  the  geometry  on  the 
sphere.  This  analogy  has  its  foundation  in  the  fact  that  the 
sphere  can  be  freely  moved  upon  itself,  so  that  propositions 
in  every  way  analogous  to  the  postulates  of  congruence  on 
the  plane  hold  for  equal  figures  on  the  sphere. 

Let  us  try  to  generalize  this  example.  In  order  that  a 
suitably  bounded  surface,  by  bending  but  without  stretching, 
can  be  moved  upon  itself  in  the  same  way  as  a  plane,  a  cer- 
tain number  \K\  invariant  with  respect  to  this  bending,  must 
have  a  constant  value  at  all  points  of  the  surface.  This  number 
was  introduced  by  Gauss  and  called  the  Curvature.'^  [In 
English  books  it  is  usually  called  Gauss's  Curvature  or  tlie 
Measure  of  Curvature.] 

I  Remembering  that  the  curvature  at  any  poir  t  of  a  plane 
curve  is  the  reciprocal  of  the  radius  of  the  osculating  circle  for 
that  point,  we  shall  now  show  that  the  curvature  at  a  point  M  of  the 
surface  can  be  defined.  Having  drawn  the  normal  n  to  the  surface 
at  M,  we  consider  the  pencil  of  planes  through  n,  and  the  corre- 
sponding pencil  of  curves  formed  by  their  intersections  with  the 
surface.  In  this  pencil  of  (plane)  curves,  there  are  two,  orthogonal 
to  each  other,  whose  curvatures,  as  defined  above,  are  maximum 
and  minimum.  The  product  of  their  curvatures  is  Gauss's  Curva- 
ture  of  the    Surface   at  M.  This  Curvature   has    one   most   marked 

9* 


I  32    ^  •  The  Later  Development  of  Non-Euclidean  Geometry. 


Surfaces  of  Constant  Curvature  can  be  actually  con- 
structed.   The  three  cases 

K^O,     A'>6>,     K<^0, 
have  to  be  distinguished. 

For  K^=  6>,  we  have  the  developable  surfaces  [applic- 
able to  the  plane]. 

For  K^  O,  we  have  the  surfaces  applicable  to  a  sphere 
of  radius  i  :  "j/ A',  and  the  sphere  can  be  taken  as  a  model 
for  these  surfaces. 

For  K<^  O,  we  have  the  surfaces  applicable  to  the 
Pseudosphere,  which  can  be  taken  as  a  model  for  the  surfaces 
of  constant  negative  curvature. 


Pseudosphere. 

Fig.  54- 

The  Pseudosphere  is  a  surface  of  revolution.  The  equat- 
ion of  its  meridian  curve  (the  tractrix  ^)  referred  to  the  axis 


characteristic.  It  is  unchanged  for  every  bending  of  the  surface 
which  does  not  involve  stretching.  Thus,  if  two  surfaces  are 
applicable  to  each  other  in  the  sense  of  the  text,  they  ought  to 
have  the  same  Gaussian  Curvature  at  corresponding  points  [Gauss], 

This  result,  the  converse  of  which  was  proved  by  Minding 
to  hold  for  vSurfaces  of  Constant  Curvature,  shows  that  surfaces, 
freely  movable  upon  themselves,  are  characterised  by  constancy  of 
curvature. 

^  The    tractrix   is   the    curve  in  which  the  distance  from   the 


Surfaces  of  Constant  Curvature. 


133 


of  rotation  z,  and  to  a  suitably  chosen  axis  of  ;c  perpendicular 
to  z,  is 


kJ^y  k^—x2 


(i)  z  =  k\og'^    I         —Vk'-x% 

where  k  is  connected  with  the  Curvature  K  by  the  equation 

To  the  pseudosphere  generated  by  (i)  can  be  applied 
any  portion  of  the  surface  of  constant  curvature  —  ,-. 


Surface  of  Constant  Negative  Curvature.^ 
Fig.  56. 


point    of    contact    of    a    tangent    to    the    point    where    it    cuts    its 
asymptote  is  constant. 

I  Fig.  56  is  reproduced  from  a  photograph  ef  a  surface  con- 
structed by  Beltrami.  The  actual  model  belongs  to  the  collection 
of  models  in   the  Mathematical  Institute  of  the  University  of  Pavia. 


I  34    V.    The  Later  Development  of  Non-Euclidean  Geometry. 

§  68.  There  is  an  analogy  between  the  geometry  on  a 
surface  of  constant  curvature  and  that  of  a  portion  of  a  plane, 
both  taken  within  suitable  boundaries.  We  can  make  this 
analogy  clear  by  tratislatiug  the  fundamental  definitions  and 
properties  of  the  one  into  those  of  the  other.  This  is  indicat- 
ed shortly  by  the  positions  which  the  corresponding  terms 
occupy  in  the  following  table: 

(a)  Surface.  (a)  Portion  of  the  plane. 

(b)  Point.  (b)  Point. 

(c)  Geodesic.  (c)  Straight  line. 

(d)  Arc  of  Geodesic.  (d)  Rectilinear  Segment. 

(e)  Linear  properties  of  the  (e)  Postulates  of  Order  for 
Geodesic.  points  on  a  Straight  Line. 

(f)  A  Geodesic  is  determined  (f)  A  Straight  Line  is  deter- 
by  two  points.  mined  by  two  points. 

(g)  Fundamental  properties  (g)  Postulates  of  Congruence 
of  the  equality  of  Geode-  for  Rectilinear  Segments 
sic  Arcs  and  Angles.  and  Angles. 

(h)  If  two  Geodesic  triangles  (h)  If  two  Rectilinear  triang- 
have  their  two  sides  and  les   have  their  two  sides 

the   contained   angles   e-  and  the  contained  angles 

qual,  then  the  remaining  equal,  then  the  remaining 

sides  and  angles  are  equal.  sides  and  angles  are  equal. 

It  follows  that  we  can  retain  as  common  to  the  geome- 
try of  the  said  surfaces  all  those  properties  concerning  bound- 
ed regions  on  a  plane,  which  in  the  Euclidean  system  are 
independent  of  the  Parallel  Postulate,  when  no  use  is  made 
of  the  complete  plaiic  [e.  g.,  of  the  infinity  of  the  straight 
line]  in  their  demonstration. 

We  must  now  proceed  to  compare  the  propositions  for 
a  bounded  region  of  the  plane,  depending  on  the  Euclidean 
hypothesis,  with  those  which  correspond  to  them  in  the  geo- 
metry on  the  surface  of  constant  curvature.  We  have,  e.  g., 
the  proposition  that  the  sum  of  the  angles  of  a  triangle  is 


Geometry  on  a  Surface  of  Constant  Curvature.  i^c 

equal  to  two  right  angles.    The  corresponding  property  does 
not  generally  hold  for  the  surface. 

Indeed  Gauss  showed  that  upon  a  surface  whose  curva- 
ture K  is  constant  or  varies  from  point  to  point,  the  surface 
integral 

over  the  whole  surface  of  a  geodesic  triangle  ABC,  is  eqtial 
io  the  excess  of  its  three  angles  over  two  right  angles.  ' 

i.  e.  \[  KdS  =A-VB+  C—  IT. 

ABC 

Let  us  apply  this  formula  to  the  surfaces  of  constant 
curvature,  distinguishing  the  three  possible  cases — 
Case  1.  K=^0. 

In  this  case  we  have 

UxdS  =  O;  that  is  ^  +  ^  +  C=tx. 

ABC 

Thus  the  sum  of  the  angles  of  a  geodesic  triangle  on  sur- 
faces of  zero  curvature  is  equal  to  two  7'ight  angles. 

Case  II.  ^=i>  ^• 

In  this  case  we  have 

ABC  ABC 

But   {^dS  =  area  of  the  triangle  ABC  =  A. 

^^=A-\-  B-\-  C—-K. 

From  this  equation  it  follows  that 
^  +  ^  +  C>  TT, 
and  that  L=k^  {A^  B  ■\-  C— tt). 


1  Cf.  BlANCHi's  work  referred  to  above;  Chapter  VI. 


I  ?5    V.  The  Later  Development  of  Non-Euclidean  Geometry. 


That  is: 

a)  The  sum  of  the  angles  of  a  geodesic  triangle  on  sur- 
faces of  constant  positive  curvature  is  greater  than  two  right 
afigles. 

b)  The  area  of  a  geodesic  triangle  is  proportional  to  the 
excess  of  the  sum  of  its  angles  over  two  right  angles. 


X--T,<0 


Case  III. 

In  this  case  we  have 

ABC  ABC 

where  we  again  denote  the  area  of  the  triangle  ABC  hy  A. 
Then  we  have 

From  this  it  follows  that 

A-ir  B  +  C<Tr, 
and  that  A  =  /è^Tt  —  ^  -  B—  C). 

That  is: 

a)  The  sum  of  the  angles  of  a  geodesic  triangle  on  sur- 
faces of  constant  negative  curvature  is  less  than  two  right  angles. 

b)  The  area  of  a  geodesic  triangle  is  proportional  to  the 
difference  between  the  sum  of  its  angles  and  two  right  angles. 

We  bring  these  results  together  in  the  following  table: 

Surfaces  of  Constant  Curvature. 


"Value  of  the  Curvature 

Model 
of  the  Surface 

Character   of  the  Triangle 

K^O                    Plane 

<^^+<^^+<^C=TT 

^'T. 

Sphere 

^^  +  ^^  +  <^C>Tr 

K^  — 


k2 


Pseudosphere 


^A-\-^B-^^C<rz 


The  Geodesic  Triangle.  1 27 

With  the  geometry  of  surfaces  of  zero  curvature  and  of 
surfaces  of  constant  positive  curvature  we  are  already  ac- 
quainted, since  they  correspond  to  Euclidean  plane  geometry 
and  to  spherical  geometry. 

The  study  of  the  surfaces  of  constant  negative  curvature 
was  begun  by  F.  Minding  [1806 — 1885]  with  the  investiga- 
tion of  the  surfaces  of  revolution  to  which  they  could  be  ap- 
plied.* The  following  remark  of  Minding's,  fully  proved 
by  D.  Codazzi  [1824 — 1873],  establishes  the  trigonometry 
of  such  surfaces.  In  the  formulae  of  spherical  trigonometry  let 
the  angles  be  kept  fixed  and  the  sides  multiplied  by  i  =  Y--i- 
Then  we  obtaifi  the  equations  which  are  satisfied  by  the  elements 
of  the  geodesic  triangles  on  the  surf  aces  of  cofistatit  negative  cur- 
vature.^ These  equations  [the  pseudospherical  trigonometryl 
evidently  coincide  with  those  found  by  Taurinus;  in  other 
words,  with  the  formulae  of  the  geometry  of  Lobatschewsky- 

BOLVAI. 

§  69.  From  the  preceding  paragraphs  it  will  be  seen  that 
the  theorems  regarding  the  sum  of  the  angles  of  a  triangle  in 
the  geometry  on  surfaces  of  constant  curvature,  are  related  to 
those  of  plane  geometry  as  follows: — 

For  K=  O  they  correspond  to  those  which  hold  on  the 
plane  in  the  case  of  the  Hypothesis  of  the  Right  Angle. 

For  K'^  O  they  correspond  to  those  which  hold  on  the 
plane  in  the  case  of  the  Hypothesis  of  the  Obtuse  Angle. 

1  Wie  sick  entscheldeii  lasst,  ob  zivei  gegebene  knimme  Flachen 
aufelnander  abwickelbar  sind  oder  tticht;  nebst  Bemerknngen  iiber  die 
Fliichen  von  unveranderlichem  Kriimtmtngsmasse.  Crelle^s  Journal, 
Bd.  XIX,  p.  370-387  (1839). 

2  Minding:  Beitrage  zur  Theorie  der  kiirzesten  Linien  aiif  krummen 
Flachen.  Crelle's  Journal,  Bd.  XX,  p.  323—327  (1^40).  D.  Codazzi: 
Intorno  alle  superficie,  le  guali  hanno  costante  il  prodotto  de'  dice  raggi 
di  curvatura.  Ann.  di  Scienze  Mat.  e  Fis.  T.  Vili,  p.  346 — 355 
O857). 


I  •^S     V.  The  Later  Development  of  Non-Euclidean  Geometry. 

For  K<CO  they  correspond  to  those  which  hold  on  the 
plane  in  the  case  of  the  Hypothesis  of  the  Acute  Angle. 

The  first  of  the  results  is  evident  a  priori,  since  we  are 
concerned  with  developable  surfaces. 

The  analogy  between  the  geometry  of  the  surfaces  of  con- 
stant negative  curvature,  for  example,  and  the  geometry  of 
LoBATSCHEWSKY-BoLYAi,  could  be  made  still  more  evident  by 
arranging  in  tabular  form  the  relations  between  the  elements 
of  the  geodesic  triangles  traced  upon  those  surfaces,  and  the 
formulse  of  Non-Euclidean  Trigonometry.  Such  a  comparison 
was  made  by  E.  Beltrami  in  his  Saggio  di  interpretazione  della 
geometria  non-euclidea.  ' 

In  this  way  it  will  be  seen  that  the  geometry  upon  a  sur- 
face of  constant  positive  or  negative  curvature  can  be  con- 
sidered as  a  concrete  interpretation  of  the  Non-Euclideati  Geo- 
inetry,  obtained  in  a  bounded  plane  area,  with  the  aid  of  the 
Hypothesis  of  the  Obtuse  Angle  or  that  of  the  Acute  Angle. 

The  possibility  of  interpreting  the  geometry  of  a  two- 
dimensional  manifold  by  means  of  ordinary  surfaces  was  ob- 
served by  B.  RiEMANN  [1826 — 1866]  in  1854,  the  year  in 
which  he  wrote  his  celebrated  memoir:  Ober  die  Hypothesen 
welche  der  Geometrie  zugrunde  liegen.^   The  developments  of 


1  Giorn.  di  Mat.,  T.  VI,  p.  284—312  (1868).  Opere  Mat., 
T.  I,  p.  374 — 405  (Hoepli,  Milan,  1902). 

2  Riemanns  iVerke,  1.  Aufl.  (1876),  p.  254 — 312:  2.  Aufl. 
(1892),  p.  272 — 287.  It  was  read  by  RlEMANN  to  the  Philosophical 
Faculty  at  Gottingen  as  his  Hahilitatioiisschrift,  before  an  audience 
not  composed  solely  of  mathematicans.  For  this  reason  it  does 
not  contain  analytical  developments,  and  the  conceptions  intro- 
duced are  mostly  of  an  intuitive  character.  Some  analytical  ex- 
planations are  to  be  found  in  the  notes  on  the  Memoir  sent  by  RiE- 
MANN  as  a  solution  of  a  problem  proposed  by  the  Paris  Academy 
[Rietnatiits  IVerke,  I,  Aufl.,  p.  384 — 391).  The  philosophical  basis 
of  the  Habilitatioiisschriji  is  the  study  of  the  properties  of  things 
from    their    behaviour    as    infinitesimals.      Cf.    Klein's    discourse! 


Beltrami  and  Riemann. 


139 


Non-Euclidean  Geometry  in  the  direction  of  Differential  Ge- 
ometry are  directly  due  to  this  memoir. 

Beltrami's  interpretation  appears  as  a  particular  case  of 
Riemann's.  It  shows  clearly,  from  the  properties  of  surfaces 
of  constant  curvature,  that  the  chain  of  deductions  from  the 
three  hypotheses  regarding  the  sum  of  the  angles  of  a  triangle 
must  lead  to  logically  consistent  systems  of  geometry. 

This  conclusion,  so  far  as  regards  the  Hypothesis  of  the 
Obtuse  Angle,  seems  to  contradict  the  theorems  of  Saccheri, 
Lambert,  and  Legendre,  which  altogether  exclude  the  possi- 
bility of  a  geometry  founded  on  that  hypothesis.  However 
the  contradiction  is  only  apparent.  It  disappears  if  we  remem- 
ber that  in  the  demonstration  of  these  theorems,  not  only 
the  fundamental  properties  of  the  bounded  plane  are  used,  but 
also  those  of  the  complete  plane,  e.  g.,  the  property  that  the 
straight  line  is  infinite. 

Principles    of  Plane  Geometry   on  the  Ideas  of 
Riemann. 

§  70.  The  preceding  observations  lead  us  to  the  foun- 
dation of  a  metrical  geometry,  which  excludes  Euclid's  Postul- 


Riemann  and  seme  BedeiUimg  in  der  Entwickelung  der  modertten 
Mathematik.  Jahresb.  d.  Deutschen  Math.  Ver.,  Bd.  IV,  p.  72 — 82 
(1894),  and  the  Italian  translation  by  E.Pascal  in  Ann.  di  Mat.,  (2), 
T.  XXIII,  p.  222.  The  Habilitationsschrift  was  first  published  in  1867 
after  the  death  of  the  author  [Gott.  Abh.  XIII]  under  the  editor- 
ship of  Dedekind.  It  was  then  translated  into  French  by  J.  HoiJEL 
[Ann.  di  Mat.  (2).  T.  Ill  (1870),  Oeuvres  de  Riemann,  (1876)];  into 
English,  by  \V.  K.  Clifford  [Nature,  Vol.  VIII,  (1873)],  and  again 
by  G.  B.  Halsted  [Tokyo  sagaku  butsurigaku  kwai  kiji,  Vol.  VII, 
(1895);  into  Polish,  by  DiCKSTEiN  (Comm.  Acad.  Litt.  Cracov. 
Vol.  IX,  1877);  into  Russian,  by  D.  Sintsoff  [Mem.  of  the  Phy- 
sical Mathematical  Society  of  the  University  of  Kasan,  (2),  Vol.  Ill, 
App.  (1893)]. 


I^O     V.  The  Later  Development  of  Non-Euclidean  Geometry. 

ate,  and  adopts  a  more  general  point  of  view  than  that  for- 
merly held  : 

(a)  We  assume  that  we  start  from  a  bounded  plane  area 
{normal  region),  and  not  from  the  whole  plane. 

(Ò)  We  regard  as  postulates  those  elementary  propositio7is, 
which  are  revealed  to  us  by  the  senses  for  the  region  originally 
taken;  the  propositions  relative  to  the  straight  line  being  determ- 
ined by  two  points,  to  congruence,  etc. 

{c)  We  assume  that  the  properties  of  the  initial  region  can 
be  extended  to  the  neighbourhood  of  any  point  on  the  plane  \jve 
do  not  say  to  the  complete  plane.,  viewed  as  a  whole]. 

The  geometry,  built  upon  these  foundations,  will  be  the 
most  general  plane  geometry,  consistent  with  the  data  which 
rigorously  express  the  result  of  our  experience.  These  results 
are,  however,  limited  to  an  accessible  region. 

From  the  remarks  in  §  69,  it  is  clear  that  the  said  geo- 
metry will  find  a  concrete  interpretation  in  that  of  the  sur- 
faces of  constant  curvature. 

This  correspondence,  however,  exists  only  from  the 
point  of  view  {differejitial)  according  to  which  only  bounded 
regions  are  compared.  If,  on  the  other  hand,  we  place  our- 
selves at  the  {integral)  point  of  view,  according  to  which  the 
geometry  of  the  whole  plane  and  the  geometry  on  the  sur- 
face are  compared,  the  correspondence  no  longer  exists.  In- 
deed, from  this  standpoint,  we  cannot  even  say  that  the  same 
geometry  will  hold  on  two  surfaces  with  the  same  constant 
curvature.  For  example,  a  circular  cylinder  has  a  constant 
curvature,  zero,  and  a  portion  of  it  can  be  applied  to  a  region 
of  a  plane,  but  the  entire  cylinder  cannot  be  applied  in  this 
way  to  the  entire  plane.  The  geometry  of  the  complete  cy- 
linder thus  differs  from  that  of  the  complete  Euclidean  plane. 
Upon  the  cylinder  there  are  closed  geodesies  (its  circular 
sections),  and,  in  general,  two  of  its  geodesies  (helices)  meet 
in  an  infinite  number  of  points,  instead  of  in  just  two. 


Riemann's  New  Geometry.  141 

Similar  differences  will  in  general  appear  between  a  me- 
trical Non-Euclidean  geometry,  founded  on  the  postulates 
enunciated  above,  and  the  geometry  on  a  corresponding  sur- 
face of  constant  curvature. 

When  we  attempt  to  consider  the  geometry  on  a  surface 
of  constant  curvature  (e.  g.,  on  the  sphere  or  pseudosphere) 
as  a  whole,  we  see,  in  general,  that  the  fundamental  property 
of  a  normal  region  that  a  geodesic  is  fully  determined  by  two 
points  ceases  to  hold.  This  fact,  however,  is  not  a  necessary 
consequence  of  the  hypotheses  on  which,  in  the  sense  above 
explained,  a  general  metrical  Non-Euclidean  geometry  of  the 
plane  is  based.  Indeed,  when  we  examine  whether  a  system 
of  plane  geometry  is  logically  possible,  which  will  satisfy  the 
conditions  (a),  (b),  and(c),  and  in  which  the  postulates  of  con- 
gruence and  that  a  straight  line  is  fully  determined  by  two 
points  are  valid  on  the  complete  plane,  we  obtain,  in  addition 
to  the  ordinary  Euclidean  system,  the  two  following  systems 
of  geometry: 

1.  The  system  of  Lobatsc/iewsky-Bolyai,  already  explain- 
ed, in  which  two  parallels  to  a  straight  line  pass  through  a, 
point. 

2.  A  netv  system  (called  Rietnann's  system)  which  cor- 
responds to  Saccheri's  Hypothesis  of  the  Obtuse  Angle,  and 
in  which  no  parallel  lines  exist. 

In  the  latter  system  the  straight  line  is  a  closed  line  of 
finite  length.  We  thus  avoid  the  contradiction  to  which  we 
would  be  led  if  we  assumed  that  the  straight  line  were  open 
(infinite).  This  hypothesis  is  required  in  proving  Euclid's  The- 
orem of  the  Exterior  Angle  [I.  1 6]  and  some  of  Saccheri's 
results. 

§  71.  RiEMANN  was  the  first  to  recognize  the  existence 
of  a  system  of  geometry  compatible  with  the  Hypothesis  of 
the  Obtuse  Angle,  since  he  was  the  first  to  substitute  for  the 


142     V.  The  Later  Development  of  Non-Euclidean  Geometry. 

hypothesis  that  the  straight  Hne  is  infinite^  the  more  general 
one  that  it  is  unbounded.  The  difterence,  which  presents  it- 
self here,  between  infinite  and  imboimded  is  most  important. 
We  quote  in  regard  to  this  Riemann's  own  words  :  * 

'In  the  extension  of  space  construction  to  the  infinitely 
great,  we  must  distinguish  between  unboundedness  and  iiifinite 
extent;  the  former  belongs  to  the  extent  relations;  the  latter  to 
the  measure  relations.  That  space  is  an  unbounded  three-fold 
manifoldness  is  an  assumption  which  is  developed  by  every 
conception  of  the  outer  world;  according  to  which  every  in- 
stant the  region  of  real  perception  is  completed  and  the  pos- 
sible positions  of  a  sought  object  are  constructed,  and  which 
by  these  applications  is  for  ever  confirming  itself.  The  un- 
boundedness of  space  possesses  in  this  way  a  greater  empiri- 
cal certainty  than  any  external  experience,  but  its  infinite  ex- 
tent by  no  means  follows  from  this;  on  the  other  hand,  if  we 
assume  independence  of  bodies  from  position,  and  therefore 
ascribe  to  space  constant  curvature,  it  must  necessarily  be 
finite,  provided  this  curvature  has  ever  so  small  a  positive 
value.' 

Finally,  the  postulate  which  gives  the  straight  line  an  in- 
finite length,  implicitly  contained  in  the  work  of  preceding 
geometers,  is  to  Riemann  as  fit  a  subject  of  discussion  as  that 
of  parallels.  What  Riemann  holds  as  beyond  discussion  is 
the  iinboimdediiess  of  space.  This  property  is  compatible  with 
the  hypothesis  that  the  straight  line  is  infinite  (open),  as  well 
as  with  the  hypothesis  that  it  is  finite  (closed). 

The  logical  possibility  of  Riemann's  system  can  be  de- 
duced from  its  concrete  interpretation  in  the  geometry  of  the 
sheaf  of  tines.    The  properties  of  the  sheaf  of  lines  are  trans- 


I  [This    quotation    is    taken    from    Clifford's    translation    in 
Nature,  referred  to  above.     (Teil  III,  S  2  of  Riemann's  Memoir.)]. 


The  Geometry  of  the  Sheaf. 


143 


lated  readily  into  those  of  Riemann's  plane,  and  vice  versa, 
with  the  aid  of  the  following  dictionary  : 


Sheaf 

Plane 

Line 

Point 

Plane  [Pencil] 

Straight  line 

Angle  between  two  Lines 

Segment 

Dihedral  Angle 

Angle 

Trihedron 

Triangle 

We  now  give,  as  an  example,  the  'translation'  of  some 
of  the  best  known  propositions  for  the  sheaf: 

a)  The  sum   of  the  three         a)  The  sum   of  the  three 


dihedral  angles  of  a  trihedron 
is  greater  than  two  right 
dihedral  angles. 

b)  All  the  planes  which  are 
perpendicular  to  another 
plane  pass  through  a  straight 
line. 

c)  With  every  plane  of 
the  sheaf  let  us  associate  the 
straight  line  in  which  the 
planes  perpendicular  to  the 
given  plane  all  intersect.  In 
this  way  we  obtain  a  corres- 
pondence between  planes  and 
straight  lines  which  enjoys 
the  following  property:  The 
straight  lines  corresponding 
to  the  planes  of  a  pencil 
[Ebenenbiischel,  set  of  planes 
through  one  line,  the  axis  of 
the   pencil]    lie   on   a  plane, 


angles  of  a  triangle  is  greater 
than  two  right  angles. 

b)  All  the  straight  lines 
perpendicular  to  another 
straight  line  pass  through  a 
point. 

c)  With  every  straight  line 
in  the  plane  let  us  associate 
the  point  in  which  the  lines 
perpendicular  to  the  given 
line  intersect.  In  this  way  we 
obtain  a  correspondence  be- 
tween lines  and  points,  which 
enjoys  the  following  pro- 
perty: 

The  points  corresponding 
to  the  lines  of  a  pencil  lie  on 
a  straight  line^  which  in  its  turn 
has  for  corresponding  point 
the  vertex  of  the  pencil. 


144     ^"  '^^^  Later  Development  of  Non-Euclidean  Geometry, 

which  in  its  turn  has  for  cor-         The   correspondence   thus 
responding   line  the   axis   of    defined  is  called  absolute  po- 
the  pencil.    The  correspond-     larity  of  the  plane, 
enee    thus  defined  is   called 
absolute  [orthogonal]  polarity 
of  the  sheaf. 

§  72.  A  remarkable  discovery  with  regard  to  the  Hypo- 
thesis of  the  Obtuse  Angle  was  made  recently  by  Dehn. 

If  we  refer  to  the  arguments  of  Saccheri  [p.  37], 
Lambert  [p.  45],  Legendre  [p.  56],  we  see  at  once  that 
these  authors,  in  their  proof  of  the  falsehood  of  the  Hypo- 
thesis of  the  Obtuse  Angle,  avail  themselves,  not  only  of  the 
hypothesis  that  the  straight  line  is  infinite,  but  also  of  the 
Archimedea?i  Hypothesis.  Now  we  might  ask  ourselves  if  this 
second  hypothesis  is  required  in  the  proof  of  this  result.  In 
other  words,  we  might  ask  ourselves  if  the  two  hypotheses, 
one  of  which  attributes  to  the  straight  line  the  character  of 
open  lines,  while  the  other  attributes  to  the  sum  of  the  angles 
of  a  triangle  a  value  greater  than  two  right  angles,  are  com- 
patible with  each  other,  when  the  Postulate  of  Archimedes  is 
excluded.  Dehn  gave  an  answer  to  this  question  in  his 
memoir  quoted  above  (p.  30),  by  the  construction  of  a  iVw/- 
Archimedean  geometry,  in  which  the  straight  line  is  open, 
and  the  sum  of  the  angles  of  a  triangle  is  greater  than  two 
right  angles.  Thus  the  second  of  Saccheri's  three  hypotheses 
is  compatible  with  the  hypothesis  of  the  open  straight  line 
in  the  sense  of  a  Non- Archimedean  system.  This  new 
geometry  was  called  by  Dehn  Noji-Legendrean  Geometry  [cf. 
S  59,  P-  121]. 

§  73.  We  have  seen  above  that  the  geometry  of  a 
surface  of  constant  curvature  (positive  or  negative)  does  not 
represent,  in  general,  the  whole  of  the  Non-Euclidean  geo- 


Hubert's  Theorem. 


145 


metry  on  the  plane  of  Lobatschewky  and  of  Riemann.  The 
question  remains  whether  such  a  correspondence  could  not 
be  effected  with  the  help  of  some  particular  surface  of  this 
nature. 

The  answer  to  this  question  is  as  follows  : 
i)  There  does  not  exist  any  regular'^  analytic  surface 
on  which  the  geometry  of  Lobatschrujsky-Bolyai  is  altogether 
valid  [Hilbert's  Theorem].^ 


1  In  other  words,  free  from  singularities. 

2  Uber  Flacheti  von  konstanter  Gatissscher  A'nimmung.  Trans. 
Amer.  Math.  Soc.  Vol.  II,  p.  86  —  99  (1901);  Grundlagen  der  Geo- 
metrie, 2.  Aufl.  p.  162 — 175.     (Leipzig,  Teubner,  1903). 

This  question,  which  Hilbert's  Theorem  answers,  was  first 
suggested  to  mathematicians  by  Beltrami's  interpretation  of  the 
LoBATSCHEWKY-BoLyAi  Geometry.  In  1870  Helmholtz— in  his 
lecture,  Uber  U}-spning  und  Bedeuhing  der  geometrischen  Axiome, 
(Vortrage  und  Reden,  Bd.  II.  Brunswick,  1844)— had  denied  the 
possibility  of  constructing  a  pseudospherical  surface,  extending 
indefinitely  in  every  direction.  Also  A.  Gennocchi — in  his  Lettre 
à  M.  Qiietelet  sur  diverses  questions  tiiathèmatiques,  [Belgique  Bull.  (2). 
T.  XXXVI,  p.  181— 198  (1873)],  and  more  fully  in  his  Memoir, 
Sur  Ulte  mhnoire  de  D.  Foncenex  et  sur  les  geometries  non-euclidieunes, 
[Torino  Memorie  (3),  T.  XXIX,  p.  365—404  (1877)],  showed  the 
insufficiency  of  some  intuitive  demonstrations,  intended  to  prove 
the  concrete  existence  of  a  surface  suitable  for  the  representation 
of  the  entire  Non -Euclidean  plane.  Also  he  insisted  upon  the 
probable  existence  of  singular  points — (as  for  example,  those  on 
the  line  of  regression  of  Fig.  54) — in  every  concrete  model  of  a 
surface  of  constant  negative  curvature. 

So  far  as  regards  Hilbert's  Theorem,  we  add  that  the 
analytic  character  of  the  surface,  assumed  by  the  author,  has  been 
shown  to  be  unnecessary.  Cf.  the  dissertation  of  G.  Lutkemeyer  : 
Uber  den  analytischen  Charakter  der  Integrale  von  partiellen  Differ  en- 
tialgleichungen,  (Gottingen,  1902).  Also  the  Note  by  E.  Holmgren: 
Sur  les  surfaces  à  courbure  constante  negative,  [Comptes  Rendus,  I  Sem., 
p.  840—843  (1902)]. 

[In  a  recent  paper  Sur  les  surfaces  à  courbure  constante  negative, 
(Bull.  Soc.  Math,  de  France,  t.  XXXVII  p.  51—58,  1909)  É.  GouRSAT 

10 


IaQ     V.  The  Later  Development  of  Non-Euclidean  Geometry. 

2)  A  surface  on  which  the  geometry  of  the  piatte  of 
Riema?in  7uould  be  altogether  valid  itiust  be  a  closed  surface. 

The  only  regular  analytic  closed  surface  of  constant  posi- 
tive curvature  is  the  sphere  [Liebmann's  Theorem].^ 

But  on  the  sphere,  in  normal  regions  of  which  Riemann's 
geometry  is  valid,  two  lines  always  meet  in  two  (opposite) 
points. 

We  therefore  conclude  that: 

In  ordinary  space  there  are  no  surfaces  7vhich  satisfy  in 
their  complete  extent  all  the  properties  of  the  Non-Euclideati 
planes. 

§  74.  At  this  place  it  is  right  to  observe  that  the  sphere, 
among  all  the  surfaces  whose  curvature  is  constant  and  different 
from  zero,  has  a  characteristic  that  brings  it  nearer  to  the 
plane  than  all  the  others.  Indeed  the  sphere  can  be  moved 
upon  itself  just  as  the  plane,  so  that  the  properties  of  con- 
gruence are  valid  not  only  for  normal  regions,  but,  as  in  the 
plane,  for  the  surface  of  the  sphere  taken  as  a  whole. 

This  fact  suggests  to  us  a  method  of  enunciating  the 
postulates  of  geometry,  which  does  not  exclude,  a  priori,  the 
possible  existence  of  a  plane  with  all  the  characteristics  of 
the  sphere,  including  that  of  opposite  points.    We  would 


has  discussed  a  problem  slightly  less  general  than  that  enunciated 
by  Hilbert,  and  has  succeeded  in  proving — in  a  fairly  simple 
manner — the  impossibility  of  constructing  an  analytical  surface  of 
constant  curvature,  which  has  no  singular  points  at  a  finite  distance.] 
I  Eiiw  nelle  EigenschaJ't  der  Kiigel,  Gott.  Nachr.  p.  44 — 54 
(1899).  This  property  is  also  proved  by  Hilbert  on  p.  172 — 175 
of  his  Gnindlagen  der  Geometrie.  We  notice  that  the  surfaces  of 
constant  positive  curvature  are  necessarily  analytic.  Cf.  I.UTKE- 
meyer's  Dissertation  referred  to  above  (p.  163),  and  the  memoir 
by  Holmgren  :  Ober  eine  K lasse  voji  partiellen  Differcntialgleickuiigen 
der  ziveiten   Ordntmg,  Math.  Ann.  Bd.  TVII,  p.  407  —  420  (1903). 


The  Elliptic  and   Spherical  Planes.  jaj 

need  to  assume  that  ilic   following  relations  were  true  for 
the  plane: 

i)  The  postulates  (/>),  (c-)  [cf.  §  70]  in  every  normal 
region. 

2)  The  postulates  of  congruence  in  the  whole  of  the 
plane. 

Thus  we  would  have  the  geometrical  systems  of  Euclid, 
of  LOBATSCHEWSKY-BoLVAi.  and  of  RiEMANN  {f/i£  elliptic  type), 
which  we  have  met  above,  where  two  straight  lines  have 
only  one  common  point  :  and  a  second  Riemann's  system 
Kthe  spherical  type),  where  two  straight  lines  have  always  two 
common  points. 

§  75-  We  cannot  be  quiie  certain  what  idea  Riemann 
had  formed  of  his  complete  plane,  whether  he  had  thought 
of  it  as  the  elliptic  p'laiu,  or  the  spherical  plane,  or  had 
recognized  the  possibility  of  both.  This  uncertainty  is  due 
to  the  fact  that  in  his  memoir  he  deals  with  Differential 
Geometry  and  devotes  only  a  lew  words  to  the  complete 
forms.  Further,  those  who  continued  his  labours  in  this  direc- 
tion, among  them  Beltrami,  always  considered  Riemann's 
geometry  in  connection  with  the  sphere.  They  were  thus  led 
to  hold  that  on  the  complete  Riemann's  plane,  as  on  the 
sphere  (owing  to  the  existeurc  of  the  opposite  ends  of  a 
diameter),  the  postulate  that  a  straight  line  is  determined  by 
two  points  had  exceptions,"  and  that  the  only  form  of  the 
plane  compatible  with  the  Hypothesis  of  the  Obtuse  Angle 
would  be  the  spherical  plane. 

Cf.  for  example,  tlic  sliort  reference  to  the  geometry  of 
space  of  constant  positive  curvature  with  which  Beltrami  concludes 
his  memoir:  Teoria  fondamt'iifale  lit'^^li  spazii  di  atrvatura  costante, 
Ann.  di  Mat.  (2).  T.  11,  p.  354 — 355  (1868);  or  the  French  trans- 
lation of  this  memoir  by  J.  llouKi.,  Ann.  So.  d.  I'Ecole  Norm.  Sup. 
T.  VI,  p.  347-377. 

10* 


148    V.    The  Later  Development  of  Non-Euclidean  Geometry. 

The  fundamental  characteristics  of  the  elliptic  plane 
were  given  by  A.  Cayley  [1821 — 1895]  in  1859,  but  the 
connection  between  these  properties  and  Non-Euclidean 
geometry  was  first  pointed  out  by  Klein  in  187  i.  To  Klein 
is  also  due  the  clear  distinction  between  the  two  geometries 
of  RiEMANN,  and  the  representation  of  the  elliptic  geometry 
by  the  geometry  of  the  sheaf  [cf  S  7i]- 

To  make  the  difference  between  the  spherical  and 
elliptic  geometries  clearer,  let  us  fix  our  attention  on  two 
classes  of  surfaces  presented  to  us  in  ordinary  space:  the 
surface  with  two  faces  {two-sided)  and  the  surface  with  one 
face  {one-sided). 

Examples  of  two-sided  surfaces  are  afforded  by  the 
ordinary  plane,  the  surfaces  of  the  second  order  (conicoidal, 
cylindrical,  and  spherical),  and  in  general  all  the  surfaces 
enclosing  solids.  On  these  it  is  possible  to  distinguish  two 
faces. 

An  example  of  a  one-sided  surface  is  given  by  the 
Leaf  of  MÒBIUS  [MoBiussche  Blatt],  which  can  be  easily 
constructed  as  follows:  Cut  a  rectangular  strip  AB  CD.  In- 
stead of  joining  the  opposite  sides  AB  and  CD  and  thus 
obtaining  a  cylindrical  surface,  let  these  sides  be  joined 
after  one  of  them,  e.  g.,  CD,  has  been  rotated  through  two 
right  angles  about  its  middle  point.  Then  what  was  the 
upper  face  of  the  rectangle,  in  the  neighbourhood  of  CD, 
is  now  succeeded  by  the  lower  face  of  the  original  rectangle. 
Thus  on  Mobius'  Leaf  the  distinction  between  the  tivo 
faces  becomes  impossible. 

If  we  wish  to  distinguish  the  one-sided  surface  from  the 
wo-sided  by  a  characteristic,  depending  only  on  the  intrinsic 
properties  of  the  surface,  we  may  proceed  thus: — We  fix  a 
point  on  the  surface,  and  a  direction  of  rotation  about  it 
Then  we  let  the  point  describe  a  closed  path  upon  the  sur- 
face, which  does  not  leave  the  surface;  for  a  two-sided  sur- 


A  One-Sided  Surface. 


149 


face  the  point  returns  to  its  initial  position  and  the  final 
direction  of  rotation  coincides  with  the  initial  one;  for  a  one- 
sided surface,  [as  can  be  easily  verified  on  the  Leaf  of  Mobius, 
when  the  path  coincides  with  the  diametral  line]  there  exist 
closed  paths  for  which  the  final  direction  of  rotation  is  oppos- 
ite to  the  initial  direction. 

Coming  back  to  the  two  Riemann's 
planes,  we  can  now  easily  state  in  what 
their  essential  difi"erence  consists  :  the  spher- 
ical plaiie  has  the  character  of  the  two-sided 
surface,  and  the  elliptic  plane  that  of  the  one- 
sided surface. 

The  property  of  the  elliptic  plane  here  ^he  Leaf  of  Mobius. 
enunciated,  as  well  as  all  its  other  propert-  '^"  ^^' 

ies,  finds  a  concrete  interpretation  in  the  sheaf  of  lines.  In 
fact,  if  one  of  the  lines  of  the  sheaf  is  turned  about  the  vertex 
through  half  a  revolution,  the  two  rotations  which  have  this 
line  for  axis  are  interchanged. 

Another  property  of  the  eUiptic  plane,  allied  to  the 
preceding,  is  this  :  The  elliptic  plane,  unlike  the  Euclidean 
plane  and  the  other  Non-Euclidean  planes,  is  not  divided  by 
its  lines  into  two  parts.  We  can  state  this  property  other- 
wise: If  two  points  A  and  A'  are  given  upon  the  plane,  and 
an  arbitrary  straight  hne,  we  can  pass  from  A  to  A'  by  a 
path  which  does  not  leave  the  plane  and  does  not  cut  the 
line.^  This  fact  is  'translated'  by  an  obvious  property  of  the 
sheaf,  which  it  would  be  superfluous  to  mention. 

§  76.  The  interpretation  of  the  spherical  plane  by  the 
sheaf  of  rays  (straight  lines  starting  from  the  vertex)  is  ana- 
logous to  that  given  above  for  the  elliptic  plane.    The  trans- 

I  A  surface  which  completely  possesses  the  properties  of  the 
elliptic  plane  was  constructed  by  W.  Boy.  [Gott.  Berichte,  p.  20 
—23  (1900);  Math.  Ann.  Bd.  LVII,  p.  151 — 184  (1903)]. 


ICQ    V.  The  Later  Developmeru  of  Non-Euclidean  Geometry. 

lation  of  the  properties  of  this  plane  into  the  properties  of 
the  sheaf  of  rays  is  effected  ])y  the  use  of  a  'dictionary' 
similar  to  that  of  §  71,  in  which  the  word  J>oÌ7ìf  is  found 
opposite  the  word  rav. 

The  comparison  of  the  sheaf  of  rays  with  the  sheaf  of 
lines  affords  a  useful  means  of  making  clear  the  connections, 
and  revealing  the  differences,  whic-h  are  to  be  found  in  the 
two  geometries  of  Rikmann. 

We  can  consider  two  sheaves,  with  the  same  vertex,  the 
one  of  lines,  the  other  of  rays.  Tt  is  clear  that  to  every  line 
of  the  first  correspond  two  ra)s  of  the  second;  that  every 
figure  of  the  first  is  formed  by  two  symmetrical  figures  of  the 
second;  and  that,  with  certain  restrictions,  the  metrical  pro- 
perties of  the  two  forms  are  the  same.  Thus  if  we  agree  to 
regard  the  two  opposite  rays  of  tlie  sheaf  of  rays  as  forming 
one  element  only,  the  sheaf  of  rays  and  the  sheaf  of  lines 
are  identical. 

The  same  considerations  :ipply  to  the  two  Riemann's 
planes.  To  every  point  of  the  elliptic  plane  correspond 
two  distinct  and  opposite  points  of  the  spherical  plane;  to 
two  lines  of  the  first,  which  pass  through  that  point,  corres- 
pond two  lines  of  the  second,  which  have  two  points  in 
common;  etc. 

The  elliptic  plane,  when  compared  with  the  spherical 
plane,  ought  to  be  regarded  as  a  tfoubh' plane. 

With  regard  to  the  elliptic  j)laue  and  the  spherical 
plane,  it  is  right  to  remark  th:it  tlii-  formulae  of  absolute  tri- 
gonometry, given  in  §  56,  can  he  applied  to  them  in  every 
suitably  bounded  region.  This  follows  from  the  fact,  al- 
ready noted  in  S  58,  that  the  formulae  of  absolute  trigonom- 
etry hold  on  the  sphere,  'and  the  geometry  of  the  sphere,  so 
far  as  regards  normal  regions,  coincides  with  that  of  these 
two  planes. 


Riemann's  Solid  Geometry.  jci 

Principles  of  Riemann's  Solid  Geometry. 

§  77.  Returning  now  to  solid  geometry,  we  start  from 
the  philosophical  foundation  that  the  postulates,  although 
we  grant  them,  by  hypothesis,  an  actual  meaning,  express 
truths  of  experience,  which  can  be  verified  only  in  a  bounded 
region.  We  also  assume,  that  on  the  foundation  of  these  postul- 
ates points  in  space  are  represented  by  three  coordinates. 

On  such  an  (analytical)  representation,  every  line  is 
given  by  three  equations  in  a  single  variable: 

and  we  must  now  proceed  to  determine  a  function  j,  of 
the  parameter  t,  which  shall  express  the  length  of  an  arc  of 
the  curve. 

On  the  strength  of  the  distributive  property,  by  which 
the  length  of  an  arc  is  equal  to  the  sum  of  the  lengths  of 
the  parts  into  which  we  imagine  it  to  be  divided,  such  a 
function  will  be  fully  determined  when  we  know  the  element 
of  distance  (ds)  between  two  infinitely  near  points,  whose 
coordinates  are 

jCi  +  dXi ,  x,  +  dx2 ,  X,  +  dxy 

RiEMANN  starts  with  very  general  hypotheses,  which 
are  satisfied  most  simply  by  assuming  that  ds',  the  square 
of  the  element  of  distance,  is  a  quadratic  expression  in- 
volving the  differentials  of  the  variables,  which  always  re- 
mains positive: 

ds'  ==  Zfly-  dxi  dxj , 
where  the  coefficients  aij  are  functions  oi  x^,  x^,  Xy 

Then,  admitting  the  principle  of  superposition  of  figures, 
it  can  be  shown  that  the  fimction  a;j  must  be  such  that,  with 
the  choice  of  a  suitable  system  of  coordinates, 

ds'=  ^ ^ 

I-U— (jri2+;<r22  4-jf32) 
4 


I  e  2     V.  The  Later  Development  of  Non-Euclidean  Geometry. 

In  this  formula  the  constant  K  is  what  Riemann,  by  an  ex- 
tension of  Gauss's  conception,  calls  the  Curvature  of  Space. 

According  as  K  is  greater  than,  equal  to,  or  less  than 
zero,  we  have  space  of  constant  positive  curvature,  space 
of  zero  curvature,  or  space  of  constant  negative  curvature. 

AVe  make  another  forward  step  when  we  assume  that  the 
principle  of  superposition  [the  principle  of  movement]  can  be 
extended  to  the  whole  of  space,  as  also  the  postulate  that  a 
straight  line  is  always  determined  by  two  points.  In  this  way 
we  obtain  three  forms  of  space;  that  is,  three  geometries 
which  are  logically  possible,  consistent  with  the  data  from 
which  we  set  out. 

The  first  of  these  geometries,  corresponding  to  positive 
curvature,  is  characterised  by  the  fact  that  Riemann's  system 
is  valid  in  every  plane.  For  this  reason  space  of  positive 
curvature  will  be  unbounded  and  finite  in  all  directions. 
The  second,  corresponding  to  zero  curvature,  is  the  ordinary 
Euclidean  geometry.  And  the  third,  which  corresponds  to 
negative  curvature,  gives  rise  in  every  plane  to  the  geometry 

of  LOBATSCHEWSKV-BOLYAI. 

The  Work  of  Helmholtz  and  the  Investigations 
of  Lie. 

§  78.  In  some  of  his  philosophical  and  mathematical 
writings,*  Helmholtz  [1821  — 1894]  has  also  dealt  with  the 


I  Uber  die  Ihatsachlichen  Gniiidlagcn  der  Geometrie,  Heidelberg, 
Verb.  d.  naturw.-med.  Vereins,  Bd.  IV,  p.  197 — 202  (1868);  Bd.  V, 
p.  31 — 32  (1869).  Wiss.  Abhandlungen  von  H.  Helmholtz,  Bd.  II, 
p.  610—617  (Leipzig,  1883).  French  translation  by  J.  HouEL  in 
Mém.  de  la  Soc.  des  Se.  Phys.  et  Nat.  de  Bordeaux,  T.  V,  (1868), 
and  also,  in  book  form,  along  with  the  Etudes  Gèométriques  of 
LOBATSCHEWSKV  and  the  Correspondance  de  Gauss  et  de  Schumacher, 
(^Paris,  Hermann,  1895). 

Uber  dii'Thatsachen,  die  der  Geometrie  zum  Gruude  lie;^en.     Cott. 


Helmholtz  and  Lie.  It2 

question  of  the  foundations  of  geometry.    Instead  of  assum- 
ing a  priori  the  form 

ds"^  =  XiZ/;-  dxi  dxj, 
as  the  expression  for  the  element  of  distance,  he  showed 
that  this  expression,  in  the  form  given  to  it  by  Riemann  for 
space  of  constant  curvature,  is  the  only  one  possible,  when, 
in  addition  to  Riemann's  hypotheses,  we  accept,  from  the 
beginning,  that  of  the  mobility  of  figures,  as  it  would  be  given 
by  the  movement  of  Rigid  Bodies. 

The  problem  of  Riemann-Helmholtz  was  carefully 
examined  by  S.  Lie  [1842 — 1899].  He  started  from  the 
fundamental  idea,  recognized  by  Klein  in  Helmholtz's 
work,  that  the  congruence  of  two  figures  signifies  that  they  are 
able  to  be  transfiormed  the  one  into  the  other,  by  means  of  a 
certain  point  transformation  in  space:  and  that  the  properties, 
in  virtue  of  which  congruefice  takes  the  logical  character  of 
equality,  depend  upon  the  fact  that  displacements  are  given  by 
a  group  of  transformations.^ 

In  this  way  the  problem  of  Riemann-Helmholtz  was 
reduced  by  Lie  to  the  following  form: 


Nachr.  Bd.  XV,  p.  193—221   (1868).     Wiss.  Abhandl.,  Bd.  II,  p.  618 

—639- 

The  Axioms  of  Geometry.  The  Academy,  Vol.  I,  p.  123 — l8i 
(1870);  Revue  des  cours  scient.,  T.  VII,  p.  498—501   (1870). 

Uber  die  Axiome  der  Geotjietrie.  Populare  wissenschaftliche  Vor- 
tràge.  Heft  3,  p.  21 — 54.  (Brunswick,  1876).  English  translation; 
Mind,  Vol.  I,  p.  301 — 321.  French  translation;  Revue  scientifique 
de  la  France  et  de  l'Étranger  (2).     T.  XII,   p,  1 197— 1207  (1877) 

Uber  den  Ur sprung ,  Sinn,  ii7id  Bedetiticng  der  geo?}tct)isc/ie?t 
Salze,  "Wiss.  Abh.  Bd.  II,  p.  640 — 660.  English  translation;  Mind, 
Vol.  n,  p.  212  —  224(1878). 

I  Cf.  Klein  :  Vergleichende  Betrachtungen  iiber  netiere  geometrische 
Forschungen,  (Erlangen,  1872);  reprinted  in  Math.  Ann.  Bd.  XLIII, 
p.  63 — 100  (1893).  Italian  translation  by  G.  Fang,  Ann.  di  Mat.  (2), 
T.  XVII,  p.  301-343  (1899)- 


I  e /I     V.  The  Later  Development  of  Non-Euclidean  Geometry. 

To  determine  all  the  continuous  groups  in  space  which, 
in  a  bounded  regio?t,  have  the  property  of  displacements. 

When  these  properties,  which  depend  upon  the  free 
mobiUty  of  Hne  and  surface  elements  through  a  point,  are 
put  in  a  suitable  form,  there  arise  three  types  of  groups, 
which    characterise    the    three    geometries   of  Euclid,    of 

LOBATSCHEWSKY-BOLYAI  and  of  RiEMANN.  ' 

Projective  Geometry  and  Non-Euclidean  Geometry. 

Subordination   of  Metrical   Geometry   to  Projective 

Geometry. 

§  79.  In  conclusion,  there  is  an  interesting  connection 
between  Projective  Geometry  and  the  three  geometrical 
systems  of  Euclid,  Lobatschewsky-Bolyai  and  Riemann. 

To  give  an  idea  of  this  last  method  of  treating  the 
question,  we  must  remember  that  Projective  Geometry,  in 
the  system  of  G.  C.Staudt  [1798 — 1867],  rests  simply  upon 
graphical  notions  on  the  relations  between  points,  lines 
and  planes.  Every  conception  of  congruence  and  movement 
[and  thus  of  measurement  etc,,]  is  systematically  banished. 
For  this  reason  Projective  Geometry,  excluding  a  certain 
group  of  postulates,  will  contain  a  more  restricted  number  of 
general  properties,  which  for  plane  figures  are  the  [projective] 
properties,  remaining  invariant  by  projection  and  section. 

However,  when  we  have  laid  the  foundations  of  Pro- 
jective Geometry  in  space,   7cie  can  introduce  into  this  system 


I  Cf.  Lie:  Theorie  der  Transjormalionsgritppcu.  Bd.  Ill,  p- 437 
— 543  (Leipzig,  1893).  In  connection  with  the  same  subject,  H. 
Poincaré,  in  his  memoir:  Sur  les  hypotheses  fondamctitanx  de  la 
gioinitrie  [Bull,  de  La  Soc.  Math,  de  France.  T.  XV,  p.  203 — 2l6 
('877)]»  solved  the  problem  of  finding  all  the  hypotheses,  which 
distinguish  the  fundamental  group  of  plane  Euclidean  Geometry 
from  the  other  transformation  groups. 


Projective  Geometry  and  Xon-Euclidean  Geometry.        jer 

the  metrical  conceptions,  as  relations  between  its  figures  and 
certain  definite  {metrical)  entities. 

Keeping  to  the  case  of  the  Euclidean  plane,  let  us  see 
what  graphical  interpretation  can  be  given  to  the  fundamental 
metrical  conceptions  of  parallelism  and  of  perpendicularity. 

To  this  end  we  must  specially  consider  the  line  at  infi?i- 
ity  of  the  plane,  and  the  absolute  involution  which  the  set  of 
orthogonal  lines  of  a  pencil  determine  upon  it.  The  double 
points  of  such  an  involution,  conjugate  imaginaries,  are 
called  the  circular  poiftts  (at  infinity),  since  they  are  common 
to  all  circles  in  the  plane  [Poncelet,  1822^]. 

On  this  understanding,  the  parallelism  of  two  lines  is 
expressed  graphically  by  the  property  which  they  possess  of 
meetifjg  in  a  point  on  the  line  at  iiifinity  :  the  perpendicularity 
of  two  lines  is  expressed  graphically  by  the  property  that 
their  points  at  infinity  are  conjugate  in  the  absolute  involution, 
that  is,  form  a  harmonic  range  with  the  circular  points. 
[Chasles,  1850.^] 

Other  metrical  properties,  which  can  be  expressed 
graphicallyj  are  those  relative  to  the  size  of  angles,  since 
every  equation 

F{A,B,  C...)=  O, 
between  the  angles  A,  B,  C,  .  .  .,  can  be  replaced  by 
^/loga    log_^    lo^._       \_ 

in  which  a,  b,  c  .  .  .  are  the  anharmonic  ratios  of  the  pencils 
formed  by  the  lines  bounding  the  angles  and  the  (imaginary) 
ines  Joining  the  angular  points  to  the  circular  points.    [La- 

GUERRE,    1853.3] 


1  Traiti  des  propriétés  projectives  des  figures.  2.  Ed.,  T.I.   Nr.  94, 
p.  48  (Paris,  G,  Villars,  1865). 

2  Traile  de  Géoméùie  supérieitre.     2.  Ed.,  Nr.  660,  p.  425  (Paris, 
G.  Villars,  1880). 

3  Sur  la  thcorie  des  foyers.     Nouv.  Ann.  T.  XII,  p.  57-  Oeuvres 
de  Laguerre.     T.  II,  p.  12—13  (Paris,  G.  Villars,   1902;. 


IC.0    V.  The  Later  Development  of  Xon-Euclidean  Geometry. 

More  generally  it  can  be  shown  that  the  congruence 
of  any  two  plane  figures  can  be  expressed  by  a  graphical 
relation  between  them,  the  line  at  infinity,  and  the  absolute 
involution.^  Also,  since  congruence  is  the  foundation  of  all 
metrical  properties,  it  follows  that  the  line  at  infinity  and  the 
absolute  involution  allow  all  the  properties  of  Euchdean 
metrical  geometry  to  be  subordinated  to  Projective  Geo- 
metry. TÀUS  the  metrical  properties  appear  in  projective  geometry, 
not  as  graphical  properties  of  the  figures  considered  in  them- 
selves, but  as  graphical  properties  with  regard  to  the  funda- 
mental metrical  entities,  made  up  of  the  line  at  infinity  and  the 
absolute  involution. 

The  complete  set  of  fundamental  metrical  entities  is 
called  the  absolute  of  the  plane  (Cayley). 

All  that  has  been  said  with  regard  to  the  j)lane  can 
naturally  be  extended  to  space.  The  fundamental  metrical 
entities  in  space,  which  allow  the  metrical  properties  to  be 
subordinated  to  the  graphical,  are  the  plane  at  infinity  and  a 
certain  polarity  {absolute polarity)  on  this  plane.  This  polar- 
ity is  given  by  the  polarity  of  the  sheaf,  in  which  every  line 
corresponds  to  a  plane  to  which  it  is  perpendicular  [cf.  §  7 1]. 
The  fundamental  conic  of  this  polarity  is  imaginary,  since 
there  are  no  real  lines  in  the  sheaf,  which  lie  on  the  corre- 
sponding perpendicular  plane.  It  can  easily  be  shown  that 
it  contains  all  the  pairs  of  circular  points,  which  belong  to 
the  different  planes  in  space,  and  that  it  appears  as  the  com- 
mon section  of  all  spheres.  From  this  property  the  name 
of  circle  at  infinity  is  given  to  this  fundamental  metrical 
entity  in  space. 


»  Cf.,  e.  g.  F.  Enriques,  Lezioni  di  Geomelria  proielliva,  2a.  Ed. 
p.  177 — 188  (Bologna,  Zanichelli,  1904).  There  is  a  German 
translation  of  the  first  edition  of  this  work  by  H.  Fleischer 
(Leipzig,   1903). 


Cayley's  Absolute.  icj 

§  80.  The  two  following  questions  naturally  arise  at 
this  stage: 

(i)  Can  projective  geometry  be  founded  upon  the  Non- 
Euclidean  hypothesis  Ì 

(ii)  If  such  a  foundatiofi  is  possible,  can  the  metrical 
properties,  as  in  the  Euclidean  case,  he  subordinated  to  the 
projective? 

To  both  these  questions  the  reply  is  in  the  affirmative. 
If  Riemann's  system  is  valid  in  space,  the  foundation  of 
projective  geometry  does  not  offer  any  difficulty,  since  those 
graphical  properties  are  immediately  verified,  which  give  rise 
to  the  ordinary  projective  geometry,  after  the  i^nproper  entities 
are  introduced.  If  the  system  of  Lobatschewsky-Bolyai  is 
valid  in  space,  we  can  also  again  lay  the  foundation  of  the 
projective  geometry,  by  introducing,  with  suitable  conventions, 
improper  or  ideal  points,  lines  a?id planes.  This  extension  will 
follow  the  same  lines  as  were  taken  in  the  Euclidean  case,  in 
completing  space  with  the  elements  at  infinity.  It  would  be 
sufficient,  for  this,  to  consider  along  with  the  proper  sheaf 
(the  set  of  lines  passing  through  a  point),  two  improper 
sheaves,  one  formed  by  all  the  lines  which  are  parallel  to  a 
given  line  in  one  direction,  the  other  by  all  the  lines  perpen- 
dicular to  a  given  plane;  also  to  introduce  improper  points, 
to  be  regarded  as  the  vertices  of  these  sheaves. 

Even  if  the  improper  points  of  a  plane  cannot  in  this 
case,  as  in  the  Euclidean,  be  assigned  to  a  straight  line  \the 
lifie  at  infinity\  yet  they  form  a  complete  region,  separated 
from  the  region  of  ordinary  points  {proper  points)  by  a  conic 
[limiting  conic,  or  conic  at  infinity].  This  conic  is  the  locus 
of  the  improper  points  determined  by  the  pencils  of  parallel 
lines. 

In  space  the  improper  points  are  separated  from  the 
proper  points   by  a  non-ruled  quadric   [limiting  qiiadric   or 


ic8    V.    The  Later  Development  of  Non-Euclidean  Geometry. 

quadric  at  injinity],  which  is  the  locus  of  the  improper  points 
determined  by  sets  of  parallel  lines. 

The  validity  of  projective  geometry  having  been  estab- 
hshed  on  the  Non-Euclidean  hypotheses  [Klein  ^],  to  obtain 
the  subordination  of  the  metrical  geometry  to  the  projective 
it  is  sufficient  to  consider,  as  in  the  Euclidean  case,  the 
fundametital  metrical  entities  {the  absolute)^  and  to  interpret  the 
metrical  properties  of  figures  as  graphical  relations  between 
them  and  these  entities.  On  the  plane  of  Lobatschewsky- 
BoLYAi  the  fundamental  metrical  entity  is  the  limiting  conic, 
which  separates  the  region  of  proper  points  from  that  of 
improper  points,  on  the  plane  of  Riemann  it  is  an  imaginary 
conic,  defined  by  the  absolute  polarity  of  the  plane  [cf.  p.  144]. 

In  the  one  case  as  well  as  in  the  other,  the  metrical 
properties  of  figures  are  all  the  graphical  properties  which 
remain  ufialtered  in  the  projective  transformatiotis^  leaving  the 
absolute  fixed. 

These  projective  transformations  constitute  the  00 ■J  dis- 
placements of  the  Non-Euchdean  plane. 

In  the  Euclidean  case  the  said  transformations,  (which 
leave  the  absolute  unaltered),  are  the  00  ^  transformations  of 
similarity,  among  which,  as  a  special  case,  are  to  be  found 
the  003  displacements. 

In  space  the  subordination  of  the  metrical  to  the  pro- 


1  The  question  of  the  independence  of  Projective  Geometry 
from  the  theory  of  parallels  is  touched  upon  lightly  by  Klein  in 
his  first  memoir:  Uber  die  sogenannte  Nicht-Euklidische  Geometrie, 
Math.  Ann.  Bd.  IV,  p.  573 — 625  (1871).  He  gives  a  fuller  treatment 
of  the  question  in  Math.  Ann.  Bd.  VI,  p.  112 — 145  (1873).  This 
question  is  discussed  at  length  in  our  Appendix  IV  p.  227. 

2  By  the  term  projective  transformation  is  understood  such  a 
transformation  as  causes  a  point  to  correspond  to  a  point,  a  line  to 
a  line,  and  a  point  and  a  line  through  it,  to  a  point  and  a  line 
through  it. 


Metrical  Properties  as  Graphical.  I  en 

jective  geometry  is  carried  out  by  means  of  the  limiting 
quadric  {the  absolute  of  space).  If  this  is  real,  we  obtain  the 
geometry  of  Lobatschewsky-Bolyai;  if  it  is  imaginary,  we 
obtain  Riemann's  elliptic  type. 

The  metrical  properties  of  figures  are  therefore  the  graph- 
ical properties  of  space  in  relation  to  its  absolute;  that  is,  the 
graphical  properties  which  remaiti  unaltered  in  all  the  project- 
ive transformatio7is  wJiich  leave  the  absolute  of  space  fixed. 

§  8i.  How  will  the  ideas  of  distance  and  of  angle  be 
expressed  with  reference  to  the  absolute? 

Take  a  system  of  homogeneous  coordinates  (ati,  x^,  x.^ 
on  the  projective  plane.  By  their  means  the  straight  line  is 
represented  by  a  linear  equation,  and  the  equation  of  the 
absolute  takes  the  form  : 

Qrj;  =  l-Uij  Xi  Xj  =   O. 

Then  the  distance  between  two  points  X  {x^,  X2,  x^), 
V  (y^ ,  ^2 ,  y^  is  expressed,  omitting  a  constant  factor,  by  the 
logarithm  of  the  anharmonic  ratio  of  the  range  consisting  of 
X,  V,  and  the  points  M,  JV,  in  which  the  line  X  Y  meets  the 
absolute. 

If  we  then  put 

Qjcy  =  ^aijxiyj, 

and  remember,  from  analytical  geometry,  that  the  anharm- 
onic ratio  of  the  four  points  X,  Y,  M,  N  is  given  by 


Q.v+^ 


the  expression  for  the  distance  D^cy  will  be  : — 
(i)  Z>.j,  =  -J  log  -^ 17    -^  --  . 

^^xy  ^^xy  ^^xx  ^^yy 

Introducing  the  inverse  circular  and  hyperboUc  functions, 


l6o    V-  The  Later  Uevelopment  of  Non-Euclidean  Geometry. 


(2) 


D^y   ==    ik    COS  ~^       . 

I  D,y  =/ècosh-^    ^     "'^   - 


(3) 


xD^y  =  ik  Sin  -^ ,- Jil- — - 

I 


The  constant  k,  which  appears  in  these  formulae,  is 
connected  with  Riemann's  Curvature  K  by  the  equation 

Similar  considerations  lead  to  the  projective  interpret- 
ation of  the  conception  of  angle.  The  atigle  between  hoo 
lines  is  proportional  to  the  logarithm  of  the  anharmo7iic  ratio 
of  the  pencil  which  they  fortn  with  the  tajigents  from  their 
point  of  intersection  to  the  absolute. 

If  we  wish  the  measure  of  the  complete  pencil  to  be 

2  TT,   as  in  the   ordinary  measurement,   we  must  take   the 

fraction    x  :  z  i   as  the  constant  multiplier.    Then  to  express 

'  analytically  the  angle  between  two   lines  u    (ui,    u^-,   u^), 

V  (z/i,  z^2j  2^3)}  we  put 

Y«„  =  Z  bij  Ui  uj . 

If  bij  is  the  CO  factor  of  the  element  aij  in  the  dis- 
criminant of  ^xxi  the  tangential  equation  of  the  absolute  is 
given  by 

and   the    angle    between   the   two    lines    by   the    following 
formulae: — 

^nv   +  Ku/  2   U/       uT" 

(l)  ^U,V^--j\Qg^ 


'nil  '  uii    '   vv 


Formulie  for  the  Angle. 


i6i 


(2) 


-<    ?/,   7'  ^=    COS  ~^   — ,- 

r      I  «j<     I  21V 

<^  2^,  e-  =    ^    cosh  -^ ^"^L  _ 

^  z/,  Z'  =  sin  -»     ^"'^  ^^'^         ^"" 


(3') 


z/,  z/ 


sinh 


'        ■  UU      '   VV 

Similar  expressions  hold  for  the  distance  between  two 
points  and  the  angle  between  two  planes,  in  the  geometry 
of  space.    We  need  only  suppose  that 

^xx     =    O,     ¥„„     =    O, 

represent  the  equations  (in  point  and  tangential  coordinates) 
of  the  absolute  of  space,  instead  of  the  absolute  of  the  plane. 
According  as  Q^x  =  C>  is  the  equation  of  a  real  quadric, 
without  generating  lines,  or  of  an  imaginary  quadric,  the 
formulae  will  refer  to  the  geometry  of  Lobatschewky-Bolyai, 
or  that  of  RiEMANN.' 

§  82.  The  preceding  formulae,  concerning  the  angles 
between  two  lines  or  planes,  contain  those  of  ordinary 
geometry  as  a  special  case.  Indeed  if,  for  simplicity,  we  take 
the  case  of  the  plane,  and  the  system  of  orthogonal  axes, 
the  tangential  equation  of  the  Euclidean  absolute  {f^e  circular 
points,  §  79)  is 

The  formula  (2'j,  when  we  insert 

becomes 


I  For  a  full  discussion  of  the  subject  of  this  and  the  pre- 
ceding sections,  see  Clebsch-Lindemann,  Vorlesungen  ilber  Geometrie, 
Bd.  II.  Th.  I,  p.  461  — et  seq.  (Leipzig,  1891). 

11 


1 62     V.  The  Later  Development  of  Non-Euclidean  Geometry. 
<fi  U,V  =  COS      '       ,  

from  which  we  have 

COS  (^^,  v)  = 


But  the  direction  cosines  of  the  line  ti  (Ui,  u^,  u^)  are 

cos  {U,  X)  =  —-=--,  COS  (Z^_>') 


so  that  this  equation  can  be  written 

COS  {u,  V)  =  /j  4  -+-  m^  7)12. 
the  ordinary  expression  for  the  angle  between  the  two  lines 
(/i  ;//i)  and  (4  m^. 

For  the  distance  between  two  points  the  argument  does 
not  proceed  so  simply,  when  the  absolute  degenerates  into 
the  circular  points.  Indeed  the  points  J/,  N,  where  the  line 
XY  intersects  the  absolute,  coincide  in  the  point  at  infinity  on 
this  line,  and  the  formula  (i)  gives  in  every  case: 

D^y  =  4  log  {M^N^XY)  =  A  log  I  =  o. 

However,  by  a  simple  artifice  we  can  obtain  the 
ordinary  formula  for  the  distance  as  the  limiting  case  of 
formula  (3). 

To  do  this  more  easily,  let  us  suppose  the  equations 
of  the  absolute  (not  degenerate),  in  point  and  line  coor- 
dinates, reduced  to  the  form  : 

Q.i^.   ==   ^Xi'^    +    ^X^^    +    X~^  =   O, 

Then^  putting 


equation  (3)  of  the  preceding  section  gives 


Euclid's  Geometry  as  a  Limiting  Case.  1 53 

D,y  =  ik  sin-^  /eA. 
Let  e  be  infinitesimal.  Omitting  terms  of  a  higher 
order,  we  can  substitute  K  €  A  for  sin~'  K  e  A  in  this  formula 
If  we  now  choose  k^  infinitely  large,  so  that  the  product 
ik  Y^  remains  finite  and  equal  to  unity  for  every  value  of  e, 
the  said  formula  becomes 

Let  €  now  tend  to  the  limit  zero.  The  tangential 
equation  of  the  absolute  becomes 

«i^  +  «2^  =  O; 
and   the  conic   degenerates  into  two  imaginary  conjugate 
points  on  the  line  u^  =  o.    The  formula  for  the  distance, 
on  putting 

takes  the  form 

which  is  the  ordinary  Euclidean  formula.   We  have  thus  ob- 
tained the  required  result. 

We  note  that  to  obtain  the  special  Euclidean  case  from 
the  general  formula  for  the  distance,  we  must  let  k^  tend  to 
infinity.  Since  Riemann's  curvature  is  given  by  - —  tj  ,  this 
affords  a  confirmation  of  the  fact  that  Riemann's  curvature 
is  zero  in  Euclidean  space. 

§  83.  The  properties  of  plane  figures  with  respect  to  a 
conic,  and  those  of  space  with  respect  to  a  quadric,  together 
constitute  projective  metrical  geometry.  This  was  first  studied 
by  Cayley,''  apart  from  its  connection  with  the  Non-Euclid- 


I  Sixth  Memoir  upon  Quantics.     Phil.  Trans.  Vol.  CXLIX,  p.  6 1 
-90  (1859).     Also  Collected  Works,  Vol.  II,  p.   561  — 592. 


104     ^'  ^^^  Later  Development  of  Non-Euclidean  Geometry. 

ean  geometries.   These  last  relations  were  discovered  and 
explained  some  years  later  by  F.  Klein.  ^ 

To  Klein  is  also  due  a  widely  used  nomenclature  for 
the  projective  metrical  geometries.  He  gives  the  name  hyper- 
bolic geometry  to  Cayley's  geometry,  when  the  absolute  is 
real  and  not  degenerate:  elliptic geoinetry,  to  that  in  which 
the  absolute  is  imaginary  and  not  degenerate:  parabolic 
geometry,  to  the  limiting  case  of  these  two.  Thus,  in  the 
remaining  articles,  we  can  use  this  nomenclature  to  describe 
the  three  geometrical  systems  of  Lobatschewsky-Bolyai,  of 
RiEMANN  (elliptic  type),  and  of  Euclid. 

Representation  of  the  Geometry  of  Lobatschewsky- 
Bolyai  on  the  Euclidean  Plane. 

§  84.  To  the  projective  interpretation  of  the  Non- 
Euclidean  measurements,  of  which  we  have  just  spoken,  may 
be  added  an  interesting  representation  which  can  be  given 
of  the  Hyperbolic  Geometry  on  the  Euclidean  plane.  To  ob- 
tain it,  we  take  on  the  plane  a  real,  not  degenerate,  conic  : 
e.  g.  a  circle.  Then  we  make  the  following  definitions,  relative 
to  this  circle  : 

Plane  ==  region  of  points  within  the  circle. 

Point  =  point  inside  the  circle. 

Straight  line  =  chord  of  the  circle. 

We  can  now  easily  verify  that  the  postulate  that  a 
straight  line  is  determined  by  two  points,  and  the  postulates 
regarding  the  properties  of  straight  lines  and  angles,  can  be 
expressed  as  relations,  which  are  always  valid,  when  the  above 
interpretations  are  given  to  these  terms. 

But  in  the  further  development  of  this  geometry  we  add 


I   Cf.    Uber    die   sogcnannte  Nichi-Euklidische  Geometrie.     Math. 
Aim.  Bd.  IV,  p.  573-625  (1871). 


Representation  on  the  Euclidean  Plane.  1 65 

to  these  the  postulates  of  congruence,  contained  in  the 
following  principle  of  displacement. 

If  we  are  given  two  points  A,  A'  on  the  plane,  and  the 
straight  lines  a,  a ,  respectively  passing  through  them,  there 
are  four  methods  of  superposing  the  plane  on  itself,  so  that 
A  and  a  coincide  respectively  with  A  and  a.  More  precisely: 
ofie  method  of  superposition  is  defined  by  taking  as  corre- 
sponding to  each  other,  one  ray  of  a  and  one  ray  of  a  ,  one 
section  of  the  plane  bounded  by  a  and  one  section  bounded 
by  a.  Two  of  these  displacements  are  dirt'ct  co?igruèncès 
and  two  converse  congruences. 

With  the  preceding  interpretations  of  the  entities,  point, 
Un;  and  plane,  the  principle  here  expressed  is  translated 
into  the  following  proposition: 

If  a  conic  {i\  g.,  a  circle)  is  given  in  a  platie,  and  two 
internal  points  A,  A'  are  taken,  as  also  two  chords  a,  a',  re- 
spectively passing  through  them,  there  are  four  projective  trans- 
formations of  the  plafie,  which  change  into  itself  the  space 
within  the  conic,  and  which  make  A  and  a  correspond  respect- 
ively to  A'  and  a  . 

To  fix  one  of  them^  it  is  sufficient  to  make  sure  that  a 
given  extremity  of  a  corresponds  to  a  given  extremity  of  a  , 
and  that  to  one  section  of  the  plane  bounded  by  a,  cor- 
responds a  definite  section  of  the  plane  bounded  by  ci .  Of 
these  four  transformations,  two  determine  on  the  conic  a 
projective  correspondence  in  the  same  sense,  and  two  a  pro- 
jective correspondence  i?i  the  opposite  sense. 

§  85.  We  shall  prove  this  proposition,  taking  for  sim- 
plicity two  distinct  conies  T,  t',  in  the  same  plane  or  other- 
wise. 

Let  M,  N  be  the  extremities  of  the  chord  a  [cf  Fig.  5  8]. 
Also  M\  N'  those  of  a  [cf  Fig.  59]. 


1 66     V.  The  Later  Development  of  Non-Euclidean  Geometry. 

Let  F,  P'  be  the  poles  of  a^  a  with  respect  to  the  two 
conies. 

On  this  understanding,  the  Hne  PA  intersects  the 
conic  T  in  two  real  and  distinct  points  i?,  S:  also  the  line 
P'  A  intersects  the  conic  t'  in  two  real  and  distinct  points 

A  projective  transformation  which  changes  t  into  t',  the 
line  a  into  a,  and  the  point  A  into  A,  will  make  the  point  P 
correspond  to  P\  and  the  hne  PA  to  the  line  P'  A. 


Fig.  59- 

Thus  this  transformation  determines  a  projective  cor- 
respondence between  the  points  of  the  two  conies,  in  which 
the  pair  of  points  M',  N'  corresponds  to  the  pair  of  points 
M,  N:  and  the  pair  of  points  R' ,  S'  to  R,  S. 

Vice  versa,  a  projective  transformation  between  the  two 
conies,  which  enjoys  this  property,  is  associated  with  a  pro- 
jective transformation  of  the  two  planes,  such  as  is  here  de- 
scribed.' 

But  if  we  consider  the  two  conies  t,  t',  we  see  that  to 


I  For  this  proof,  and  the  theorems  of  Projective  Geometry 
upon  which  it  is  founded,  see  Chapter  X,  p.  251  —  253  of  the  work 
of  Enriques  referred  to  on  p.  156. 


Projective  Transformations. 


167 


the  points  of  the  range  MNRS  on  T  may  be  made  to  cor- 
respond the  points  of  any  one  of  the  following  ranges  on  t': 
M'N'R'S' 

n'm's'e: 
m'n's'jr: 

N'M'R'S'. 
In  this  way  we  prove  the  existence  of  the  four  project- 
ive transformations  of  which  we  have  spoken  in  the  propos- 
ition just  enunciated. 

If  we  suppose  that  the  two  conies  coincide,  we  do  not 
need  to  change  the 
preceding  argument  in 
any  way.  We  add,  how-  p 
ever,  that  of  the  four 
transformations  only 
one  makes  the  segment 
AM  correspond  to  the 
segment  A'M\  if  at  the 
same  time  the  shaded 
parts  of  the  figure  cor- 
respond to  each  other. 

Further  the  two  transformations  defined  by  the  ranges 

/     MNRS 

\  M'lYR'S' 

determine  projections  in  the  same  sense,  while  the  other  two, 
defined  by  the  ranges  : 

f    MNRS    \         /     MNRS    \ 

\  MN'S'R'  )         \  N'M'R'S'  ) 
determine  projections  in  the  opposite  setise. 


\         /    MNRS    \ 
)'  ),        \  N'M'S'K  ) 


§  86.  With  these  remarks,  we  now  return  to  complete 
the  definitions  of  S  84,  relative  to  a  circle  given  on  the 
plane. 

Flane  =  region  of  points  within  the  circle. 


l68    V.  The  Later  Development  of  Non-Euclidean  Geometry. 

Point  =  point  within  the  circle. 

Straight  Line  =  chord  of  the  circle. 

Displacements  ==  projective  transformations  of  the  plane 
which  change  the  space  within  the  circle  into  itself. 

Semi-Revolutions  =  homographic  transformations  of  the 
circle. 

Congn/ent  Figures  =  figures  which  can  be  transformed 
the  one  into  the  other  by  means  of  the  projective  trans- 
formations named  above. 

The  preceding  arguments  permit  us  to  affirm  at  once 
that  all  the  propositions  of  elementary  plane  geometry,  asso- 
ciated with  the  concepts  straight  line,  angle  and  congruence, 
can  be  readily  translated  into  proj)erties  relative  to  the 
system  of  points  inside  the  circle,  which  we  denote  by  {S). 
In  particular  let  us  see  what  corresponds  in  {S)  to  two  per- 
pendicular lines  in  the  ordinary  plane. 

To  this  end  we  note  that  if  r,  s  are  two  perpendicular 
lines,  a  semi-revolution  of  the  plane  about  j  will  superpose 
r  upon  itself,  exchanging,  however,  the  two  rays  in  which  it 
is  divided  by  s. 

According  to  the  above  definitions,  a  semi-revolution  in 
{S)  is  a  homographic  transformation,  which  has  for  axis  a 
chord  s  of  the  circle  and  for  centre  the  pole  of  the  chord. 
The  lines  which  are  unchanged  in  this  transformation,  in  ad- 
dition to  s,  are  the  lines  passing  through  its  centre.  Thus 
in  the  system  (S)  we  must  call  two  lines  perpendicular,  when 
they  are  conjugate  with  respect  to  the  fundamcjital  circle. 

We  could  easily  verify  in  {S)  all  the  propositions  on 
perpendicular  lines.  In  particular,  that  if  we  draw  the  (imag- 
inary) tangents  to  the  fundamental  circle  from  the  common 
point  of  two  conjugate  chords  in  (^),  these  tangents  form 
a  harmonic  pencil  with  the  perpendicular  lines  [cf.  p.  155].' 

I  This   representation   of  the  Non-Euclidean   plane   has  been 


The  Distance  between  two  Points.  i5q 

§  87.  Let  US  now  see  how  the  distance  between  two 
points  can  be  expressed  in  this  conventional  measurement, 
which  is  being  taken  for  the  interior  of  the  circle. 

To  this  end  we  introduce  a  system  of  orthogonal  coord- 
inates {x,  y),  with  origin  at  the  centre  of  the  circle. 

The  distance  between  two  points  A  {x,  j),  B  {x ,  y) 
in  the  plane  with  which  we  are  dealing  cannot  be  represen- 
ted by  the  usual  formula 

Y{x~xy\{y-y)\ 

since  it  is  not  invariant  for  the  projective  transformations 
which  we  have  called  displacements.  The  distance  must  be  a 
function  of  the  coordinates,  invariant  for  the  said  transforma- 
tions, which  for  points  on  the  straight  line  possesses  the  dis- 
tributive property  given  by  the  formula 

dist.  {AE)  =  dist.  {AC)  -f  dist.  {CB). 

Now  the  anharmonic  ratio  of  the  four  points  A^  B,  M, 
N,  where  M,  N  are  the  extremities  of  the  chord  AB,  is  a 
relation  between  the  coordinates  {x,  y),  {x\  y')  of  AB, 
remaining  invariant  for  all  projective  transformations  which 
leave  the  ~_  fundamental  circle  fixed.  The  most  general  ex- 
pression, possessing  this  invariant  property,  will  be  an  arbi- 
trary function  of  this  anharmonic  ratio. 

If  we  remember  that  the  said  function  must  be  distrib- 
utive in  the  sense  above  indicated,  we  must  assume  that, 
except  for  a  multiplier,  it  is  equal  to  the  logarithm  of  the 
anharmonic  ratio, 

(ABM^T)  =  ^^: -^^,- 

We  shall  thus  have 

distance  (AB)  =  ^  log  {ABMN). 


employed   by  Grossmann    in    carrying   out    a    number   of  the  con- 
structions  of  Non-Euclidean  Geometry.     Cf.  Appendix,  III,  p.  225. 


170    V-  The  Later  Development  of  Non-Euclidean  Geometry. 

In  a  similar  way  we  proceed  to  find  the  proper  ex- 
pression for  the  angle  between  two  straight  lines.    In  this  case 

we  must  notice  that  if  we  wish  the  right  angle  to  be  ex- 
it 
pressed  by  — ,  we  must  take  as  constant  multiplier  of  the 

logarithm  the  factor  1:22. 

Then  we  shall  have  for  the  angle  between  a  and  b, 

^^>^=^  2/  ^^^  iabmn), 

where  m,  n  are  the  conjugate  imaginary  tangents  from  the 
vertex  of  the  angle  to  the  circle,  and  {a  b  m  n)  is  the  an- 
harmonic  ratio  of  the  four  lines  a,  b,  in  and  «,  expressed 
analytically  by 

sin  ia  ni)     sin  {a  n) 
sin   [d  m)  '  sin   (i  n) 

%  88.  A  glance  at  what  was  said  above  on  the  sub- 
ordination of  the  metrical  to  the  projective  geometry  (S  81) 
will  show  clearly  that  the  preceding  formulge^  regarding  the 
distance  and  angle,  agree  with  those  which  we  would  have  in 
the  Non-Euclidean  plane,  if  the  absolute  were  a  circle.  This 
would  be  sufficient  to  suggest  that  the  geometry  of  the  system 
(6')  gives  a  concrete  representation  of  the  geometry  of 
LoBATSCHEWSKY-BoLVAi.  However,  as  we  wish  to  discuss 
this  point  more  fully,  let  us  see  how  the  definition  and  pro- 
perty of  parallels  are  translated  in  \S). 

Let  r  (z^i,  U2,  u^)  and  r  {i\,  V2,  v^  be  two  difterent 
chords  of  the  fundamental  circle. 

Let  the  circle  be  referred  to  an  orthogonal  Cartesian 
set  of  axes,  with  the  centre  for  origin,  and  let  us  take  the 
radius  as  unit  of  length. 

Then  we  have 

x^  -Vy^  —  1=0, 
u^-\-v^  —  1=0, 
for  the  point  and  line  equation  of  the  circle. 


The  Angle  between  two  Lines. 


171 


Making  these  equations  homogeneous,  we  obtain 

Xj,^  +  x,'  —  x^'  =  O, 

Ui'+U2'  —   //3^  =  O. 

The  angle  ^r,  r  between  the  two  straight  hnes  r  and 
r  can  be  calculated  by  means  of  the  formula  (3')  of  §  81, 
if  we  put 


^uu  '-'■  u,'  +  u. 


u. 


3  ' 


We  thus  obtain 


sin  <5C  r,  r 


V  (U1V3 ViUiY —  (2^22^3 — "v-^ò^  —  iu-^-i^ V-^U^^' 


But  the  lines  r,  r  are  given  by 

Xt_H^-\- X-Jl^-V  X.jLl.^  =  O, 
XtJ)-!^  +  .^22^2  +  -^3^3  ^=  O  ; 

and  they  meet  in  the  point, 

x^  =  u^v^,  —  U^^2  , 

X^  ==  U^^x—-U{U^, 
X^  ==  U1V2  —  2/2  Z'l- 

Thus  the  preceding  expression  for  this  angle  takes 
the  form 

/    \  •        V  '  '     \Xx  Xx  X^,   ) 

(4)       sm  <^r,r=     ^  —     " . 

l/(2<!.^  +  «2^  —  2^3^)  (Z'l^  +  Z^a^  — t'3^) 

From  this  it  is  evident  that  the  necessary  and  sufficient 
condition  that  the  angle  be  zero  is  that  the  numerator  of 
this  fraction  should  vanish. 

Now  if  this  numerator  is  zero,  the  point  {x-^,  x^,  x.^,  in 
which  the  chords  intersect,  must  lie  on  the  circumference  of 
the  fundamental  circle,  and  vice  versa  (Fig.  61). 

Therefore  in  our  Ì7iterpretation  of  the  geometrical  pro- 
positions by  77ieans  of  the  system  (S),  we  must  call  two  chords 
parallel,  when  they  iueet  in  a  point  on  the  circumference  of  the 


J  72    V.  The  Later  Development  of  Non-Euclidean  Geometry. 


fundatnental  circle,  since  the  angle  between  those  two  chords 
is  zero. 

Since  there  are  two  chords  through  any  point  within  a 
circle  which  join  this  point  to  the  ends  of  any  given  chord, 
the  fundamental  proposition  of  hyperbolic  geometry  will  be 
verified  for  the  system  {S). 

§  89.  We  proceed  to  find  for  the  system  {S)  the 
formula  regarding  the  angle  of  parallelism.  To  do  this  we 
first  calculate  the  angle  OMN,  between  the  axis  of_j'  and 
the  line  MN,  joining  a  point  M  on  the  axis  of^  to  the  ex- 
tremity of  the  axis  oi  x  (Fig.  62). 


Fig.  61. 


Fig.  62. 


Denoting  by  a  the  ordinary  distance  of  the  two  points 
M  and  O,  the  homogeneous  coordinates  of  the  line  MN  and 
the  line  OM  axQ,  respectively  {a,  i,  —  a),  (i,  O,  o)  and  the 
coordinates  of  their  common  point  are  (o,  a,  1). 

Then  from  (4)  of  the  preceding  article, 
sin  <^  OMN  =  \^i-a2. 

On  the  other  hand,  the  distance;,  according  to  our  con- 
vention, between  the  two  points  O  and  M  is  given  by  (2)  of 
S  81  as 

OM  =  h  cosh  -'  -—- — 

Thus 

OM  _         I 


cosh  ~  = 


The  Angle  of  Parallelism.  172 


Comparing  these  two  results,  we  have 

,    OM  I 

cosh  —r-  = 


k  sin  <^  OMN'> 

a  relation  which  agrees  with  that  given  by  TaurinuS;  Lo- 
BATSCHEWSKY  and  BoLYAi  for  the  angle  of  parallehsm  [cf. 
p.  90]. 

§  90.  We  proceed,  finally,  to  see  how  the  distance  be- 
tween two  neighbouring  points  {the  element  of  distance)  is 
expressed  in  the  system  (vS),  so  that  we  may  be  able  to 
compare  this  representation  of  the  hyperbolic  geometry  with 
that  given  by  Beltrami  [cf.  g  69]. 

Let  {x^y)^  (x  +  dx,y  +  dy)  be  two  neighbouring  points. 
Their  distance  ds  is  calculated  by  means  of  (2)  of  §  81  if  we 
substitute  : 

Qxz  =  x'+y^—  I, 

Qj,y  =  (x  +  dxy  +  iy  +  dyY—  i, 

^xy  =  X  {x  +  dx)  -^-y  (y  +  dy)  —  i . 

Since  the  angle  is  small,  we  may  substitute  the  sine  for 
the  angle,  and  we  have 

_    ,2  (dx^  +  dy»){l—x2  —y2)  4-  [xdx  +ydyy 

(X2   +-y2  _    I)     ((;,  ^  dxY   +    fy  +  dyY  -    I)) 

Thus,  omitting  terms  higher  than  the  second  order, 
we  have 

^^.  _  ^2       {dx^  4-  dy2)  (I  —  x2  —y2)  +  ^xdx  +ydyy 

(l  X2  —y2)2 

or 

(c)      ds"-  =  k"        (^  —y^)  ^-^^  -^ixydxdy^  {l—x2)dy2  ^ 

-^  {l—X2—y2)2 

Now  we  recall  that  Beltrami,  in  i868,  interpreted  the 
geometry  of  Lobatschewsky-Bolvai  by  that  on  the  surfaces 
of  constant  negative  curvature.  The  study  of  the  geometry 
on  such  surfaces  depends  upon  the  use  of  a  system  of  coord- 
inates on  the  surface,  and  the  law  according  to  which  the 
element  of  distance  {ds)  is  measured.    The  choice  of  a  suitable 


I  HA    R.  The  Later  Development  of  Non-Euclidean  Geometry. 

system  (?/,  v)  enabled  Beltrami  to  put  the  square  of  ds  in 
this  form: 

(I  —  v^')  dit^  -f-  zitvdudri  -\-(\  —  n^) dv^ 
k , 

(I   «2  —  2/2)2 

where  the  constant  k^  is  the  reciprocal,  with  its  sign  changed, 
of  the  curvature  of  the  surface.' 

In  studying  the  properties  of  these  surfaces  and  in  mak- 
ing a  comparison  between  them  and  the  metrical  results  of 
the  geometry  of  Lobatschewsky-Bolyai,  Beltrami  in  his 
classical  memoir,  quoted  on  p.  138,  employed  the  following 
artifice: 

He  represented  the  points  of  the  surface  on  an  aux- 
iliary plane,  such  that  the  point  {u,  z')  of  the  surface  corre- 
sponded to  the  point  on  the  plane  whose  Cartesian  coord- 
inates (x,}>)  were  {u,  v).  The  points  on  the  surface  were 
then  represented  by  points  inside  the  circle 

x^  +y^  —  I  =  O; 

the  points  at  infinity  on  the  surface  by  points  on  the  cir- 
cumference of  the  circle:  its  geodesies  by  chords:  parallel 
geodesies  by  chords  meeting  in  a  point  on  the  circumference 
of  the  said  circle.  Then  the  expression  for  {dsY  took  the 
same  form  as  that  given  in  (5),  which  states  the  form  to  be 
used  for  the  element  of  distance  in  the  system  {S). 

It  follows  that,  by  his  representation  of  the  surfaces  of 
constant  negative  curvature  on  a  plane,  Beltrami  was 
led  to  one  of  the  projective  metrical  geometries  of  Cayley, 
and  precisely  to  the  metrical  geometry  relative  to  a  funda- 
mental circle,  given  above  in  §§  80,  81. 

I  Risoluzione  del  problema  di  riportare  i  punti  di  una  superficie 
sop>ra  un  piano  in  modo  che  le  linee  geodetiche  vengano  rappresentate 
da  linee  rette.  Ann.  di  Mat.  T.  VII,  p.  185 — 204  (1866).  Also 
Opere  Matematiche.     T.  I,  p.  262 — 280  (Milan,   1902). 


Beltrami's  Geometry  and  Projective  Geometry.  jyc 

§  gi.  The  representation  of  plane  hyperbolical  geo- 
metry on  theEudidean  plane  is  capable  of  being  extended  to 
the  case  of  solid  geometry.  To  represent  the  solid  geometry 
of  LoBATSCHEWSKY-BoLYAi  in  Ordinary  space  we  need  only 
adopt  the  following  definitions  for  the  latter: 

Space  =  Region  of  points  inside  a  sphere. 

I^owt  =  Point  inside  the  sphere. 

Straight  Line  =  Chord  of  the  sphere. 

Plane  =  Points  of  a  plane  of  section  which  are  inside 
the  sphere. 

Displacements  =  Projective  transformations  of  space, 
which  change  the  region  of  the  points  inside  the 
sphere  into  itself,  etc. 

With  this  'Dictionary'  the  propositions  of  hyperbolic 
solid  geometry  can  be  translated  into  corresponding  proper- 
ties of  the  Euclidean  space,  relative  to  the  system  of  points 
inside  the  sphere.' 

Representation  of  Riemann's  Elliptic  Geometry  in 
Euclidean  Space. 

§  92.  So  far  as  regards  plane  geometry,  we  have  already 
remarked  [pp.  142 — 3]  that  the  geometry  of  the  ordinary 
sheaf  of  lines  gives  a  concrete  interpretation  of  the  elliptical 
system  of  Riemann.  Therefore,  if  we  cut  the  sheaf  by  an 
ordinary  plane,  completed  by  the  line  at  infinity,  we  obtain 
a  representation  on  the  Euclidean  plane  of  the  said  Rie- 
mann's plane. 


I  Beltrami  considers  the  interpretation  of  Non-Euclidean  Solid 
Geometry,  and,  in  general,  of  the  geometries  of  manifolds  of 
higher  order  in  space  of  constant  curvature,  in  his  memoir:  Teoria 
fondamentale  degli  spazii  di  curvatura  costante.  Ann.  di  Mat.  (2), 
T,  II,  p.  232—255  (1868).  Opere  Mat.  T.  I,  p.  406—429  (Milan, 
1902). 


176    V.  The  Later  Development  of  Non-Euclidean  Geometry. 

If  we  wish  a  representation  of  the  elliptic  space  in  the 
Euclidean  space,  we  need  only  assume  in  this  a  single-valued 
polarity,  to  which  corresponds  an  imaginary  quadric,  not 
degenerate.  We  must  then  take,  with  respect  to  this  quadric, 
a  system  of  definitions  analogous  to  those  indicated  above 
in  the  hyperbolic  case.  We  do  not  pursue  this  point  further, 
as  it  offers  no  fresh  difficulty. 

However  we  remark  that  in  this  representation  all  the 
points  of  the  Euclidean  space,  including  the  points  on  the  plane 
at  infinity,  would  have  a  one-one  correspondence  with  the  points 
of  Rietnann' s  space. 

Foundation  of  Geometry  upon  Descriptive 
Properties. 

§  93.  The  principles  explained  in  the  preceding  sections 
lead  to  a  new  order  of  ideas  in  which  the  descriptive  propert- 
ies appear  as  the  first  foundations  of  geometry,  instead  of 
congruence  and  displacement,  of  which  Riemann  and  Helm- 
HOLTZ  availed  themselves.  We  note  that,  if  we  do  not  wish 
to  introduce  at  the  beginning  any  hypothesis  on  the  inter- 
section of  coplanar  straight  lines,  we  must  start  from  a 
suitable  system  of  postulates,  valid  in  a  boic7ided  region  of 
space,  and  that  we  must  complete  the  initial  region  later  by 
means  oi  improper  points,  lines  and  planes  [cf.  p.  157].^ 

When  projective  geometry  has  been  developed,  the 
metrical  properties  can  be  introduced  into  space,  by  adding 
to  the  initial  postulates  those  referring  to  displacement  or 


I  For  such  developments,  cf.  Klein,  Ioc.  cit.  p.  158:  Pasch, 
Vorlesungen  iiber  neuere  Geometrie,  (Leipzig,  l882)j  SCHUR,  Uber  die 
Einfichrting  der  sogenannten  idea! en  Elemenie  in  die  projective  Geometrie, 
Math.  Ann.  Bd.  XXXIX,  p.  113 — 124  (1891):  Bonola,  Suila  intro- 
duzione degli  elementi  improprii  in  geometria  proiettiva.  Giornale  di 
Mat.  T.  XXXVIII,  p.   105— 116  (1900). 


Foundation  of  Geometry  upon  Descriptive  Properties.      177 

congruence.  By  so  doing  we  find  that  a  certain  polarity  of 
space,  allied  to  the  metrical  conceptions,  becomes  trans- 
formed into  itself  by  all  displacements.  Then  it  is  shown 
that  the  fundamental  quadric  of  this  polarity  can  only  be: 

a)  A  real,  non-ruled  quadric; 

b)  An  imaginary  quadric  (with  real  equation); 

c)  A  degetiérate  quadric. 

Thus  the  three  geometrical  systems,  which  Riemann  and 
Helmholtz  reached  from  the  conception  of  the  element  of 
distance,  are  to  be  found  also  in  this  way.* 

The  Impossibility  of  proving  Euclid's  Postulate. 

§  94.  Before  we  bring  to  a  close  this  historical  treat- 
ment of  our  subject  it  seems  advisable  to  say  a  few  words 
on  the  impossibility  of  demonstrating  Euclid's  Postulate. 

The  very  fact  that  the  innumerable  attempts  made  to 
obtain  a  proof  did  not  lead  to  the  wished-for  result,  would 
suggest  the  thought  that  its  demonstration  is  impossible.  In- 
deed our  geometrical  instinct  seems  to  afford  us  evidence 
that  a  proposition,  seemingly  so  simple,  if  it  is  provable, 
ought  to  be  proved  by  an  argument  of  equal  simplicity.  But 
such  considerations  cannot  be  held  to  afford  a  proof  of  the 
impossibility  in  question. 

If  we  put  Euclid's  Postulate  aside,  following  the  devel- 
opments of  Gauss,  Lobatschewsky  and  Bolyai,  we  can 
construct  a  geometrical  system  in  which  no  contradictions 
are  met.  This  seems  to  prove  the  logical  possibility  of  the 
Non-Euclidean  hypothesis,  and  that  Euclid's  Postulate  is 
independent  of.  the  first  principles  of  geometry  and  therefore 
cannot  be  demonstrated.    However  the  fact  that  contradictions 


I  For  the  proof  of  this  result  see  BONOLA,  Determinazione 
per  via  geometrica  dei  ire  tipi  de  spazio;  iperbolico,  parabolico,  ellittico. 
Rend.  Gire.  Mat.  Palermo,  T.  XV,  p.  56—65  (1901). 

12 


J  73    V.  The  Later  Development  of  Non-Euclidean  Geometry. 

have  not  been  met  is  not  sufficient  to  prove  this;  we  must 
be  certain  that,  proceeding  on  the  same  Hnes,  such  con- 
tradictions could  never  be  met.  This  conviction  can  be 
gained  with  absolute  certainty  from  the  consideration  of  the 
formulae  of  Non-Euclidean  geometry.  If  we  take  the  system 
of  all  the  sets  of  three  numbers  (x,  y,  z),  and  agree  to  con- 
sider each  set  as  an  analytical  point,  we  can  define  the 
distance  between  two  such  analytical  points  by  the  formulae 
of  the  said  Non-Euclidean  Trigonometry.  In  this  way  we 
construct  an  analytical  system,  which  offers  a  conventional 
interpretation  of  the  Non-Euclidean  geometry,  and  thus 
demonstrates  its  logical  possibility. 

In  this  sense  the  formulae  of  the  Non- Euclidean  Trigon- 
ometry of  Lobatscheiusky-Bolyai  give  the  proof  of  the  independ- 
ence of  Euclid's  Postulate  from  the  first  principles  of  geometry 
(regarding  the  straight  line,  the  plane  and  congruence). 

We  can  seek  a  geometrical  proof  of  the  said  independ- 
ence, on  the  lines  of  the  later  developments  of  which  we 
have  given  an  account.  For  this  it  is  necessary  to  start  from 
the  principle  that  the  conceptions,  derived  from  our  intu- 
ition, independently  of  the  correspondence  which  they  find 
in  the  external  world,  are  a  priori  logically  possible;  and  that 
thus  the  Euclidean  geometry  is  logically  possible  and  every 
set  of  deductions  founded  upon  it. 

But  the  interpretation  which  the  Non-Euclidean  plane 
hyperbolic  geometry  finds  in  the  geometry  on  the  surfaces 
of  constant  negative  curvature,  offers,  up  to  a  certain  point, 
a  first  proof  of  the  im.possibility  of  demonstrating  the  Eu- 
clidean postulate.  To  put  the  matter  in  more  exact  terms: 
by  this  means  it  is  established  that  the  said  postulate  cannot 
be  demonstrated  on  the  foundation  of  the  first  principles  of 
geometry,  held  valid  in  a  bounded  region  of  the  plane.  In 
fact,  every  contradiction,  which  would  arise  from  the 
other  postulate,    would  be  translated   into   a  contradiction 


Euclid's  Postulate  cannot  be  Proved. 


1/9 


in  the  geometry  on  the  surfaces  of  constant  negative  curv- 
ature. 

However,  since  the  comparison  between  the  hyperbolic 
plane  and  the  surfaces  of  constant  negative  curvature,  exists, 
as  we  have  seen,  only  for  bounded  regions^  we  have  not  thus 
excluded  the  possibility  that  the  Euclidean  postulate  might 
be  proved  for  the  complete  plane. 

To  remove  this  uncertainty,  it  would  be  necessary  to 
refer  to  the  abstract  manifold  of  constant  curvature,  since  no 
concrete  surface  exists  in  ordinary  space,  in  which  the  ^(?w- 
//<?/<?  hyperbolic  geometry  holds  [cf.  §  73]. 

But,  even  so,  the  impossibility  of  proving  Euclid's  Pos- 
tulate would  have  been  shown  only  for  pla7ie  geometry.  There 
would  still  remain  the  question  of  the  possibility  of  proving 
it  by  means  of  the  considerations  of  solid  geometry. 

The  foundation  of  geometry,  on  Riemann's  principles, 
whereby  the  ideas  of  the  geometry  on  a  surface  are  extended 
to  a  tliree-dimensional  region,  gives  the  complete  proof  of  the 
impossibility  of  this  demonstration.  This  proof  depends  on 
the  existence  of  a  Non-Euclidean  analytical  system.  Thus  we 
are  brought  to  another  analytical  proof.  The  same  remark 
applies  also  to  the  investigations  of  Helmholtz  and  Lie, 
though  it  might  be  argued  that  the  latter  also  offer  a  geomet- 
rical proof,  from  the  existence  of  transformation  groups  of 
the  Euclidean  space,  similar  to  the  groups  of  displacements  of 
the  Non-Euclidean  geometry.  Of  course,  it  must  be  under- 
stood that  we  here  consider  geometry  in  its  fullest  sense. 

But  the  proof  of  the  impossibility  of  demonstrating  Eu- 
clid's Postulate^  which  is  based  upon  the  projective  measure- 
ments of  Cayley,  is  simpler  and  easier  to  follow  geometrically. 

This  proof  depends  upon  the  representation  of  the 
Non-Euclidean  geometry  by  the  conventional  measurement 
relative  to  a  circle  or  to  a  sphere,  an  interpretation  which  we 


I  So    ^-  The  Later  Development  of  Non-Euclidean  Geometry. 

have  developed  at  length  in  the  case  of  the  plane   [§§  84 
—92]. 

Further  the  proof  of  the  logical  possibility  of  Riemann's 
elliptic  hypothesis  can  be  just  as  easily  derived  from  these 
projective  measurements.  For  the  plane,  the  interpretation 
which  we  have  given  of  it  as  the  geometry  of  the  sheat 
will  be  sufficient  [§  71]/ 


I  Another  neat  and  simple  proof  of  the  independence  of  the 
Fifth  Postulate  is  to  be  found  in  the  representation  of  the  Non- 
Euclidean  plane,  employed  by  Klein  and  Poincaré.  In  this  the 
points  of  the  Non-Euclidean  plane  appear  as  points  of  the  upper 
portion  of  the  Euclidean  plane,  and  the  straight  lines  of  the  Non- 
Euclidean  plane  as  semicircles,  perpendicular  to  the  straight  bound- 
ary of  this  halfplane;  etc.  The  Elliptic  Geometry  can  be  repres- 
ented in  a  similar  way;  and  the  Hyperbolic  and  Elliptic  Solid 
Geometries  can  also  be  brought  into  correspondence  with  the 
Euclidean  Space.  An  account  of  these  representations  is  to  be 
found  in  "Weber  und  Wellstein's  Encyklopàdie  der  Elemetttar- 
Mathematik,  Bd.  II  S  9— n»  P-  39 — 81  (Leipzig,  I905)  and  in 
Chapter  II  of  the  NUhi-Euklidische  Geometrie  by  H.  Liebmann 
(Sammlung  Schubert,  49,  Leipzig,  1905). 

In  Appendix  V  of  this  volume  a  similar  argument  is  given, 
based  upon  the  discussion  in  Weber-Wellstein's  volume.  Points 
upon  the  Non-Euclidean  plane  are  represented  by  pairs  of  points 
inverse  to  a  fixed  circle  on  the  Euclidean  plane;  and  straight 
lines  upon  the  one,  are  circles  orthogonal  to  the  fixed  circle  on 
the  other. 


Appendix  I. 

The  Fundamental  Principles  of  Statics  and 
Euclid's  Postulate. 


On  the  Principle  of  the  Lever. 

§  I.  To  demonstrate  the  Principle  of  the  Lever,  Archi- 
medes [287 — 212]  avails  himself  of  several  hypotheses,  some 
expressed  and  others  imphed.  Among  the  hypotheses 
passed  over  in  silence,  in  addition  to  that  which  we  would 
now  call  the  hypothesis  of  increased  constraint  ',  there  is  one 
which  definitely  concerns  the  equilibrium  of  the  lever,  and 
can  be  expressed  as  follows: 

When  a  lever  is  suspended  fro7n  its  middle  point,  it  is  in 
equilibrium,  if  a  weight  2  F  is  applied  at  one  end,  a?id  at  the 
other  another  lever  is  hung  by  its  ntiddle  point,  each  of  its  ends 
supportifig  a  weight  P} 

We  shall  not  discuss  the  various  criticisms  upon  Archi- 
medes' use  of  this  hypothesis,  nor  the  different  attempts  made 
to  prove  it.^    In  this  connection  we  shall  refer  only  to  the 


1  This  hypothesis  can  be  enunciated  as  follows:  If  several  bodies, 
subjected  to  constraints,  are  in  eqziilibritim  under  the  action  of  given 
forces,   they   will  still  be  ifz   equilibrium,   if  new   constraints  are  added 

to  those  already  in  existence.     Ci.,  for  example,    J.  Andrade,    Legons 
de  Méca7iique  Physique,  p.  59  (Paris,   1 898). 

2  Cf.  Archimedis  opera  omnia:  critical  edition  by  J.  L.  HeiberG; 
Bd.  II,  p.   142  et  seq.  (Leipzig,   1881). 

3  Cf,,    for    example,    E.   Mach,    Die   Mechanik    in    ihrer  Ent- 


1 82  Appendix  I.     The  Fundamental  Principles  of  Statics  etc. 

arguments  of  Lagrange,  since  these  will  show,  clearly  and 
simply,  the  important  link  between  this  hypothesis  and  the 
Parallel  Postulate. 


§  2.  Let  ABD  be  an  isosceles  triangle  {AD  =■  £D), 
from  whose  angular  points  A  and  £  are  suspended  two 
(cf  Fig.  63)  equal  weights  P,  while  a  weight  equal  to  2P  is 
suspended  from  D. 

This  triangle  will  be  in  equilibrium 
about  the  straight  line  MN,  joining 
the  middle  points  of  the  equal  sides, 
since  each  of  these  sides  may  be 
regarded  as  a  lever  from  whose  ex- 
tremities equal  weights  are  hung. 

But  the  equilibrium  of  the  figure 
will  also  be  secured,  if  the  triangle 
rests  upon  a  line  passing  through 
i^  the  vertex  Z)  and  the  middle  point 
C  of  the  side  AB.  Therefore,  if  E 
is  the  common  point  of  CD  and  MN, 
the  triangle  will  be  in  equilibrium,  when  suspended  from  E. 

'Or',  continues  Lagrange,  'comme  I'axe  [MN]  passe 
par  le  milieu  des  deux  cótés  du  triangle,  il  passera  aussi 
nécessairement  par  le  milieu  de  la  droite  menée  du  sommet 
du  triangle  au  milieu  [CJ  de  sa  base;  done  le  levier  trans- 
versal [CZ>]  aura  le  point  d'appui  [E]  dans  le  miheu  et 
devra,  par  consequent,  étre  charge  également  aux  bouts 
[C,  D\.  done  la  charge  que  supporte  le  point  d'appui  du 
levier;  qui  fait  la  base  du  triangle,  et  qui  est  charge,  à  ses 


ivickelung,  (3.  Aufl.,  Leipzig,  15^97);  English  translation  by  T.  J.  Mc- 
CoRMACK  (Open  Court  Publishing  Co.  Chicago,  1902).  Also,  for 
the  different  hypotheses  from  which  the  proof  of  the  principle  of 
the  lever,  can  be  obtained,  see  P.  Duhem,  Les  origines  de  la  stati- 
qiie,  (Paris,  1905),  especially  Appendix  C,  Sur  les  divers  axiomes 
d'ou  se  peut  déduire  la  ihcorie  du  levier. 


Statical  Hypothesis  equivalent  to  Postulate  V.  i8^ 

deux  extrémités  de  poids  égaux,  sera  égale  au  poids  double 
du  sommet  et,  par  consequent,  égale  à  la  somme  des  deux 
poids.*  ^ 

§  3.  Lagrange's  argument  contains  implicitly  some 
hypotheses  of  a  statical  nature,  regarding  symmetry,  addition 
of  constraints,^  etc.;  and,  in  addition,  it  involves  a  geometrical 
property  of  the  Euclidean  triangle.  But  if  we  wish  to  omit 
the  latter,  a  course  which  for  certain  reasons  seems  natural, 
the  preceding  conclusions  will  be  modified. 

Indeed,  though  we  may  still  assume  that  the  triangle 
ABD  is  in  equilibrium  about  the  point  E^  where  the  lines 
MN  and  CD  intersect,  we  cannot  assert  that  E  is  the  middle 
point  of  CD,  as  this  would  be  equivalent  to  assuming 
Euclid's  Postulate.  Consequently,  we  cannot  assert  that  the 
single  weight  2  P,  applied  at  C,  can  be  substituted  for  the  two 
weights  at  A  and  B,  since,  if  such  a  change  could  take  place, 
a  lever  would  be  in  equilibrium,  with  equal  weights  at  its  ends, 
about  a  point  which  cannot  be  its  middle  point. 

Vice  versa,  if  we  assume,  with  Archimedes,  that  two 
equal  weights  at  the  end  can  be  replaced  by  a  double 
weight  at  the  middle  point  of  the  lever,  then  we  can  easily 
deduce  that  E  is  the  middle  point  of  CD,  and  from  this  it 
will  follow  that  ABD  is  a  Euclidean  triangle. 

Hence  we  have  established  the  equivalence  of  Euclid's 
Fifth  Postulate  and  the  said  hypothesis  of  Archimedes.  Such 
equivalence  is,  of  course,  relative  to  the  system  of  hypotheses 
which  comprises,  on  the  one  hand,  the  above-named  statical 
hypotheses,  and,  on  the  other,  the  ordinary  geometrical 
hypotheses. 


1  Oeuvres  de  Lagrange,  T.  XI,  p.  4 — 5. 

2  For  an  analysis  of  \!as.  physical  principles  on  which  ordinary 
statics  is  founded,  cf.  F.  Enriques,  Problemi  della  Scienza.  Cap.  V. 
(Bologna,   1906).     German  translation,  (Leipzig,  1910). 


184   Appendix  I.     The  Fundamental  Principles  of  Statics  etc. 

With  the  modern  notation,  we  can  speak  of  forces, 
of  the  composition  of  forces,  oi  resultants,  m'ìXtz.à.  oi  weights, 
levers,  etc. 

Then  the  hypothesis  referred  to  takes  the  following 
form: 

The  resultant  of  two  equal  forces  in  the  same  plane,  applied 
at  right  angles  to  the  extremities  of  a  straight  line  and  towards 
the  same  side  of  it,  is  a  single  force  at  the  middle  point  of  the 
line,  of  double  the  intensity  of  the  given  forces. 

From  what  we  have  said  above,  if  this  law  for  the  com- 
position of  forces  were  true,  it  would  follow  that  the  ord- 
inary theory  of  parallels  holds  in  space. 

On  the  Composition *of  Forces  Acting  at  a  Point. 

§  4.  The  other  fundamental  principle  of  statics,  the 
law  of  the  Parallelogram  of  Forces,  from  the  usual  geom- 
etrical interpretation  which  it  receives,  is  closely  connected 
with  the  Euclidean  nature  of  space.  However,  if  we  examine 
the  essential  part  of  this  principle,  namely,  the  analytical 
expression  for  the  resultant  R  of  two  equal  forces  P,  acting 
at  a  point,  it  is  easy  to  show  that  it  exists  independently  of  any 
hypothesis  on  parallels. 

This  can  be  made  clear  by  deducing  the  formula 
R  =  a/'  cos  a, 
where  2  a  is  the  angle  formed  by  the  two  concurrent  forces 
from  the  following  principles: 

i)  Two  or  more  forces,  acting  at  the  same  point,  have 
a  definite  resultant. 

2)  The  resultant  of  two  equal  and  opposite  forces 
is  zero. 

3)  The  resultant  of  two  or  more  forces,  acting  at  a 
point,  along  the  same  straight  line,  is  a  force  through  the 
same  point,  equal  to  the  sum  of  tlie  given  forces,  and  along 
the  same  line. 


Composition  of  Concurrent  Forces. 


I8: 


4)  The  resultant  of  two  equal  forces,  acting  at  the  same 
point,  is  directed  along  the  line  bisecting  the  angle  between 
the  two  forces. 

5)  The  magnitude  of  the  resultant  is  a  continuous  funct- 
ion of  the  magnitude  of  the  components. 

Let  us  see  briefly  how  we  establish  our  theorem.  The 
value  i?  of  the  resultant  of  two  forces  of  equal  magnitude  /*, 
enclosing  the  angle  2  a,  is  a  function  of  P  and  a  only. 

Thus  we  can  Avrite 

i?=  2/(P,a). 

A  first  application  of  the  principles  named  above  shows 
that  R  is  proportional  to  P,  and  this  result  is  independent 
of  any  hypothesis  on  parallels  [cf  note  i,  p.  195].  Thus  the 
preceding  equation  can  be  written  more  simply  as 

R  ==    2P/{0.). 

We  now  proceed  to  find  the  form  of/"  (a). 


§  5.    Let  us  calculate  /(a)  for  some  particular  value 
of  the  angle. 
(I)  Let  a  =  45°- 

At  the  point  O  at  which  act    1  p  Q 

the  two  forces  Ft,,  P2,  of  equal 
magnitude  P,  let  us  imagine  two 
equal  and  opposite  forces  applied, 
perpendicular  to  R  and  of  magni- 

tude  —  (cf.  Fig.  64). 

At  the  same  time  let  us  imag- 
ine R  decomposed  into  two  others, 
directed  along  R  and  of  magni- 
tude 


R 


We  can  then  regard  each  force  F  as  the  resultant  of 
two  forces  at  right  angles,  of  magnitude  — . 


J  86    Appendix  I.     The  Fundamental  Principles  of  Statics  etc. 

We  thus  have 

Z'  =  2  .  ^  ./(45°). 

On  the  other  hand^  R  being  the  resultant  of  i^i  and  Pa, 
we  have 

R==  2  /y(45°)- 
From  these  two  equations  we  obtain 

/(45°)  =  \  V^' 
(II)  Again  let  a  =  60°. 

In  this  case  apply  a.t  O  a.  force  R'  equal  and  opposite 
to  R  (cf.  Fig.  65).  The  system  of  the  two  forces  R  and  of 
R'  is  in  equilibrium. 


Thus  by  symmetry,  R'  =  P. 
Therefore,  R  =  F. 
But,  on  the  other  hand, 

i?=  2  /y(6o"). 

Therefore/ (60")  =  y. 

(Ill)  Again  let  a  =  36°. 

At  O  let  the  five  forces  P^,  P^-.Pc^,  of  magnitude  P^  be 


Special  Cases.  1 87 

applied,  such  that  each  of  them  forms  with  the  next  an  angle 
of  72°  (cf.  Fig.  66). 

This  system  is  in  equilibrium. 

For  the  resultant  R  of  P2  and  P^,  we  have 

R=  2/y(36°). 
For  the  resultant  i?'  of  /'i  and  P^ ,  we  have 
R'  =  2Pf{U'). 

On  the  other  hand,  R  has  the  same  direction  as  Pc^  ; 
that  is,  a  direction  opposite  to  that  of  R. 

Therefore  2  /yCsó")  =  2  i'/(72°)  +  P. 

(i)  Therefore  2/(36°)  =  2/(72°)  +  i. 

If,  instead,  we  take  the  resultants  of  P^  and  P^ ,  and  of 
P^  and  P^,  we  obtain  two  forces  of  magnitude  2  P/  (36°), 
containing  an  angle  of  144°. 

Taking  the  resultant  of  these  two,  we  obtain  a  new 
force  R"  of  magnitude 

4  ^7(36°)/ (7  2°). 
Now  R",  by  the  symmetry  of  the  figure,  has  the  same 
line  of  action  as  P^ ,  but  acts  in  the  opposite  direction. 
Thus,  since  equihbrium  must  exist, 

i^=4/'/(36°)/(72°). 
(2)  Therefore  i  =  4/ (36°)/ (72°). 

From  the  two  equations  (i)  and  (2)  we  obtain 

/(36o)_ltV:5_/(;.o)^-f^^, 
4  4 

on  solving  for/ (36°)  and/ (7 2°). 

§  6.  By  arguments  similar  to  those  used  in  the  pre- 
ceding section  we  could  deduce  other  values  for  /  (a). 
However,  if  we  restrict  ourselves  only  to  those  just  found, 


1 88    Appendix  I.     The  Fundamental  Principles  of  Statics  etc. 


and  compare  them  with  the  corresponding  values  of  cos  a, 
we  obtain  the  following  table: 

cos  0°  =  1  /(0°)  =  I 


cos  36° 


cos  45^ 


I  +  Vs' 

4 

2 


cos  60°  =  — 


/(36°)  = 

/(45°)  = 
/(6o°)  = 


4 


COS  72°  = 


-I  +  /5 


/(72°) 


+  Vi 


/(90-)  =  o. 

This  table  suggests  the 
identity  of  the  two  functions 
y(a)  and  cos  a.  For  fuller 
p  confirmation  of  this  fact,  we 
determine  the  functional 
equation  which  _/  (a)  satis- 
fy 2  fies  (cf.  Fig.  67). 

To  this  end  let  us  con- 
sider   four    forces  F^,   P2, 
F.^,    P^    of  magnitude    P, 
acting  at  one  point,  forming 
with  each  other  the  following  angles 

-^  p,p,  =  <:  p^^p,  =  2  p 

^P,P,=-  2(a-P) 
-^  P,P,  ==  2  (a  +  P). 
We  shall  determine  the  resultant  P  of  these  four  forces 
in  two  different  ways. 

Taking  Pt_  with  P^ ,  and  P^  with  F^  we  obtain  two  forces 
i?i  and  i?j,  of  magnitude 


The  General  Case.  i8q 

inclined  at  an  angle  2  p.  Taking  the  resultant  of  Rt_  and  R2, 
we  have  a  force  li,  such  that 

i?  =  4-/y(a)/(P). 
On  the  other  hand,  taking  /'i  with  /'_,,  and  F^  with  F^^, 
we  obtain  two  resultants,  both  along  the  direction  of  R,  and 
of  magnitudes 

2Ff{a  +  ^\2Ff{a-^), 
respectively. 

These  two  forces  have  a  resultant  equal  to  their  sum, 
and  thus 

F  =  2^/(a  +  p)  +  2i'/(a— p). 

Comparing  the  two  values  of  i?,  we  find  that 
(i)  2/(a)/(P)  =/(a  +  p)  +/(a-P) 

is  the  functional  equation  required. 

If  we  now  remember  that 

cos  (a  +  P)  +  cos  (a  —  P)  =  2  cos  a  cos  P, 

and  take  account  of  the  identity  between  f  (a)  and  cos  a  in 
the  preceding  table  for  certain  values  of  a,  and  the  hy- 
pothesis that  f  (a)  is  continuous,  without  further  argument 
we  can  write 

/  (a)  =  cos  a. 
It  follows  that 

F  =  2  F  cos  a. 

The  validity  of  this  formula  of  the  Euclidean  space  is 
thus  also  established  for  the  Non-Euclidean  spaces. 

§  7.  The  law  of  composition  of  two  equal  concurrent 
forces  leads  to  the  solution  of  the  general  problem  of  the 
resultant,  since  we  can  assign,  without  any  further  hypothesis, 
the  components  of  a  force  F  along  two  rectangular  axes 
through  its  point  of  application  O. 


J  go    Appendix  I.     The  Fundamental  Principles  of  Statics  etc. 


Let  the  two  perpendicular  lines  be  taken  as  the  axes 
of  X  and  y,  and  let  i?  make  the  angles  a,  P  with  them 

Through  O  draw  the  line 
which  makes  an  angle  a  with 
Ox  and  an  angle  P  with  Oy. 
Imagine  two  equal  and  oppos- 
ite forces  Pi  and  Pz  to  act 
along  this  line  at  O,  their  mag- 
nitude  being  — .  Also  imagine 
the  force  7?  replaced  by  the 
two  equal  forces  P,  of  magni- 
tude  — ,  actmg  m  the  same 
direction  as  P. 
Then  the  system  P^,  P^,  P,  Pha.s  R  for  resultant.    But 

Pi  and  P,  taken  together,  have  a  resultant 
X  =  P  cos  a 

along  Ox:  and  P2  and  P,  taken  together,  have  a  resultant 

Y=  Rcoi  p 
along  Oy. 

These  two  forces  are  the  components  of  P  along  the 
two  perpendicular  lines.  As  to  their  magnitudes,  they  are 
identical  with  what  we  would  obtain  in  the  ordinary  theory 
founded  upon  the  principle  of  the  Parallelogram  of  Forces. 
However,  the  lines  OX  and  O  V,  which  represent  the  com- 
ponents upon  the  axes,  are  not  necessmily  the  projections  of  R, 
as  in  the  Euclidean  case.  Indeed  we  can  easily  see  that,  if 
these  lines  were  the  orthogonal  projections  of  R  upon  the 
axes,  the  Euclidean  Hypothesis  would  hold  in  the  plane. 

§  8.  The  functional  method  applied  in  S  6  to  the 
composition  of  two  equal  forces  acting  at  a  point,  is  derived 
from  D.  DE  FoNCENEx  [1734 — 1799]-     r>y  a  method  ana- 


Rectangular  Components  of  a  Fece.  Iqi 

logous  to  that  which  led  us  to  the  equation  for  /  (a)  (=  y), 
FoNCENEX  arrived  at  the  differential  equation' 

P  +  ^y=^  o. 

From  this,  on  integrating  and  taking  account  of  the  initial 
conditions  of  the  problem,  he  obtained  the  known  expression 
for/ (a). 

However  the  application  of  the  principles  of  the  In- 
finitesimal Calculus,  requires  the  continuity  and  differentiabil- 
ity of/  (a),  conditions,  which,  as  Foncenex  remarks,  involve 
the  (physical)  nature  of  the  problem.  But  as  he  wishes  to 
go  'jusqu'aux  difficultes  les  moins  fondees',  he  avails  himself 
of  the  Calculus  of  Finite  Differences,  and  of  a  Difference 
Equation,  which  allows  him  to  obtain  /  (a)  for  all  values  of 
a  which  are  commensurable  with  it.  The  case  a  incom- 
mensurable is  treated  'par  une  méthode  famiUère  aux  Géo- 
mètres  et  frequente  surtout  le  écrits  des  Anciens';  that  is,  by 
the  Method  of  Exhaustion.^ 

All  Foncenex'  argument,  and  therefore  that  given  in 


1  We  could  obtain  this  equation  from  (l)  p.  189  as  follows: 
Put  p  =  a'a  and  suppose  that /(a)  can  be  expanded  by  Taylor's 
Series  for  every  value  of  a. 

Then  we  have 

2/(a)  (/ (o)  +  '/a  /'  (o)  +  'l^  f"  (o)  .  .  .^ 

=  2/(a)  +  2  ^-/"  {«)  +  .. 

Equating    the    coefficients    of    do^    and   putting  y  = /(a)  and  k'i 

=  — /"  (o),  we  have 

d2y 

— il  4-  ^2^  =  o. 

da' 

2  Cf.  Foncenex  :  Si/r  les  prindpes  /ondameittatix  de  la  Mecan- 
ique.  Misc.  Taurinensia.  T.  II,  p.  305 — 315  (1760 — 1761).  His 
argument  is  repeated  and  explained  by  A.  GENOCCm  in  his  paper: 
Sur  un  Mémoire  de  Daviet  de  Foncenex  et  sur  les  geometries  non- 
euclidiennes.     Torino,  Memorie  (2),  T.  XXIX,  p.  366 — 371  (1877). 


IQ2    Appendix  I.     The  Fundamental  Principles  of  Statics  etc. 

§  6,  is  independent  of  Euclid's  Postulate.  However,  it 
should  be  remarked  that  Foncenex'  aim  was  not  to  make 
the  law  of  composition  of  concurrent  forces  independent  of 
the  theory  of  parallels,  but  rather  to  prove  the  law  itself. 
Probably  he  held,  as  other  geometers  [D.  Bernouilli, 
D'Alembert],  that  it  was  a  truth  independent  of  any  ex- 
perimental foundation. 

Non-Euclidean  Statics. 

§  9.  Having  thus  shown  that  the  analytical  law  for 
the  composition  of  concurrent  forces  does  not  depend  on 
Euclid's  Fifth  Postulate,  we  proceed  to  deduce  the  law  accord- 
ing to  which  forces  perpendicular  to  a  line  will  be  composed. 

Let  A,  A  be  the  points  of  application  of  two  lorces 
Pi,  P2  of  equal  magnitude  P  (cf  Fig.  69). 


Let  C  be  the  middle  point  of  AA,  and  B  a  point  on 
the  perpendicular  BC  to  AA. 

Joining  AB  and  AB,  and  putting 

<^  BAC  =  a,     <^  ABC  =  p, 
it  is  clear  that  the  force  P^  can  be  regarded  as  a  component 
of  a  force  T-s,,  acting  at  A  and  along  BA. 
The  magnitude  of  this  force  is  given  by 

P 
sin  a 


r=  -. — 


Equal  Forces  perpendicular  to  a  Line.  ig^ 

The  other  component  Q^,  at  right  angles  to  P^,  is 
given  by 

Q  =  T'cos  a  =  /'cot  a. 

Repeating  this  process  with  the  force  F2 ,  we  obtain  the 
following  system  of  coplanar  forces  : 
(i)  System  F^,  F^- 

(2)  System/',,  P,,  Q,,  Q,. 

(3)  System  7;,  T^. 

If  we  assume  that  we  can  move  the  point  of  application 
of  a  force  along  its  line  of  action,  it  is  clear  that  the  first  two 
systems  are  equivalent,  and  because  (2)  is  equivalent  to  (3), 
we  can  substitute  for  the  two  forces  jPi  ,  P2,  the  two  forces 
7;  and  7;. 

The  latter,  being  moved  along  their  lines  of  action  to  B, 
can  be  composed  into  one  force 

P  =  2rcosp  =  2/'^- 
^  sin  a 

This,  in  its  turn,  can  be  moved  to  C,  its  direction  per- 
pendicular to  A  A  remaining  unchanged. 

This  result,  which  is  obviously  independent  of  Euclid's 
Postulate,  can  be  applied  to  the  three  systems  of  geometry: 

Euclid^s  Geometry. 

In  the  triangle  ABC  we  have 

cos  P  =  sin  a. 
Therefore 

R=  2  P. 

Geometry  of  Lobatschewsky-Bolyai. 
In  the  triangle  ABC,  if  we  denote  the  side  AA  by  2  b, 
we  have 

cos  p  ^    ^    /  \ 

-. ==  cosh -r  (p.  II 7). 

sin  a  k   ^^         '^ 

Thus 

Ò 


i?  ==•  2  jP  cosh   , 


1^ 


1 94      Appendix  I.     The  Fundamental  Principles  of  Statics  etc. 

Riemann's  Geometry. 

In  the  same  triansjle  we  have 


Therefore 


cos  6  h 

-. =    COS  -r-  • 

sin  a  /C 

R  =    2  P  COS  — 


Conclusioti. 

It  is  only  in  EucUdean  space  that  the  resultant  of  two 
equal  forces,  perpendicular  to  the  same  line,  is  equal  to  the 
sum  of  the  two  given  forces.  In  the  Non-Euclidean  spaces 
the  resultant  depends,  in  the  manner  indicated  above,  on 
the  distance  between  the  points  at  which  the  two  forces  are 
applied.^ 

§  IO.  The  case  of  two  unequal  forces  P^  Q,  per- 
pendicular to  the  same  straight  line,  is  treated  in  a  similar 
manner. 

In  the  Euclidean  Geometry  we  obtain  the  known  results; 
R^  P  -V   (2, 
R     _  P  _  Q 
p-\- 1       q        P 
In  the  Geometry  of  Lobatschewsky-Bolyai  the  problem 
of  the  resultant  leads  to  the  following  equations: 

R  =  P  cosh  y  +  <2  cosh  y, 

R         _       P Q 

sinh     T  sinh  -r        sinh  -7- 

k  K  K 

Then,  by  the  usual  substitution  of  the  circular  functions 
for  the  hyperboHc,  we  obtain  the  corresponding  result  for 
Riemann's  Geometry: 


I  For  a  fuller  treatment  of  Non-Euclidean  Statics,  the  reader 
is  referred  to  the  following  authors:  J.  M.  de  Tilly,  Etudes  de 
Mécafiique  abstraiie,  Mém.  couronnés  et  autres  mém.,  T.  XXI  (1870). 
J.  Andrade,  La  Statique  et  les  Géo??iétries  de  Lobatscheivsky ,  d'Euclide, 
et  de  Riemann.     Appendix  (II)  of  the  work  quoted  on  p.  181. 


Unequal  Forces.  igc 

7?  ==  Z'  COS  y  +   (2  COS  -|-, 
R  P  Q 


■    p-\-  9  •      <]  ■     P 

sm  — T—        sin  -T-        sm  -r 

k  k  k 

In  these  formulce  /,  q,  denote  the  distances  of  the 
points  of  application  of  P  and  Q  from  that  of  R. 

These  results  can  be  summed  up  in  a  single  formula, 
valid  for  Absolute  Geometry; 

R  =  F.EP+  Q.  Eq, 
R       _    -P  __    Q 

07/ +7)  ~0(^)~Ò(?)' 

To  obtain  these  results  directly,  it  is  sufficient  to  use  the 
formulas  of  Absolute  Trigonometry,  instead  of  the  Euclidean 
or  Non-Euclidean,  in  the  argument  of  which  a  sketch  has 
just  been  given. 

Deduction  of  Plane  Trigonometry  from  Statics. 

§  II.  Let  us  see,  in  conclusion,  how  it  is  possible  to 
treat  the  converse  (\\XQsi\on:  given  the  law  of  composition  of 
forces,  to  deduce  the  fundamental  equations  of  trigonometry. 

To  this  end  we  note  that  the  magnitude  of  the  resultant 
R  of  two  equal  forces'  F,  perpendicular  to  a  line  A  A'  of 
length  2  b,  will  in  general  be  a  function  of  P  and  b. 

Denoting  this  function  by 

cp  {P,  b), 


we  have 

or  more  simply^ 


if  =  cp  (P,  b), 
R  =  P(?{b). 


I  The  proportionality  of  R  and  P  follows  from  the  laiu  of 
association  on  which  the  composition  of  forces  depends.  In  fact, 
let    us    imagine    each    of  the   forces  P,    acting   at  A  and  A',    to  be 


Iq6      Appendix  I.     The  Fundamental  Principles  of  Statics  etc. 

On  the  other  hand  in  §  9  (p.  193),  we  were  brought  to 
the  following  expression  for  J^: 

sm  a 

Eliminating  i?  and  J^,  between  these,  we  have 

/7\        cos  p 
op  (Ò)  =  -  -*-  • 
^  ^  ^         sin  a 

Thus  if  the  analytical  expression  for  (p  (/;)  is  known, 
this  formula  will  supply  a  relation  between  the  sides  and 
angles  of  a  right-angled  triangle. 

To  determine  qp  (fi),  it  is  necessary  to  establish  the 
corresponding  functional  equation. 

With  this  view,  let  us  apply  perpendicularly  to  the  line 
AA',  the  four  equal  forces  J^j,  F^,  P^,,  P^,,  in  such  a  way  that 
the  points  of  application  of  jP^  and  F^,  F^  and  jP,,  are 
distant  2  {a-\-b)  and  2  {b  —  a),  respectively  (cf.  Fig.  70). 

We  can  determine  the  resultant  R  of  these  four  forces 
in  two  different  ways: 

(i)  Taking  F,_  with  F2,  and  F^^  with  F^,  we  obtain  two 
forces  Ri,  R^  of  magnitude: 

F^{ay, 

replaced    by    n    equal    forces,    applied    at   A   and   A'.      Combining 
these,  we  would  have  for  R  the  expression 

y?  =  «  cp  (^,  b\. 
Comparing  this  result  with  the  equation  given  in  the  text,  we  have 

Similarly  we  have 

cp  (kP,  b)-^k(^{P,  b), 
for   every   rational    value  of  /c;    and  the  formula  may  be  extendeii 
to  irrational  values. 

Then  putting  P=  i   and  k  =  P  v^t  obtain 
9  [P,  Ò)  =  P(^  (6).         Q.  E.  D. 


Deduction  of  Trigonometry  from  Statics. 


197 


and  taking  R^,  R2  together,  we  obtain 
R  =  F(^  {a)  qp  {ù). 

(ii)  Taking  F^  with  F^ ,  we  obtain  a  force  of  magnitude  : 
F(p{è  +  a), 
and  taking  F2  with  F^,  we  obtain  another  of  magnitude: 
F(p(^  —  a). 
Taking  these  two  together  we  have,  finally, 
R  =  F(i>(ò  +  a)  +  Fcp(ù—a). 


A 


r-a- 


0 


b~a 


R. 


p. 


p. 


R- 


R 

Fig.  70. 


From  the  two  expressions  for  R  we  obtain  the  functional 
equation  which  qp  (^)  satisfies,  namely, 

(2)  cp(ò)  (p{a)  =  cp(l>  +  a)  +  cp  (i>  —  a). 

This  equation,  if  we  put  cp  {ò)  =  2  fib),  is  identical 
with  that  met  in  §  6  (p.  189),  in  treating  the  composition  of 
concurrent  forces. 

The  method  followed  in  finding  (2)  is  due  to  D'Alem- 
bert.^ However,  if  we  suppose  a  and  b  equal  to  each  other, 
and  if  we  note  that  qp  io)  =  2,  the  equation  reduces  to 

(3)  [9(^)]'  =  qp  (2:r)  +  2. 

This  last  equation  was  obtained  previously  by  Foncenex, 
in  connection  with  the  equilibrium  of  the  lever.^ 


1  Opuscules  mathématiqiies,  T.  VI,  p.  371  (1779). 

2  Cf.  p.  319—322    of   the    work    by    FOxNCENEX,    referred    to 


above. 


Iq8      Appendix  I.     The  Fundamental  Principles  of  Statics  etc. 

§  12.  The  statical  problem  of  the  composition  of 
forces  is  thus  reduced  to  the  integration  of  a  functional 
equation. 

FoNCENEX,  who  was  the  first  to  treat  it  in  this  way^, 
thought  that  the  only  solution  of  (3),  was  cp  (x)  =  const.  If 
this  were  so,  the  constant  would  be  2,  as  is  easily  verified. 

Later  Laplace  and  D'Alembert  integrated  (3),  obtaining 

cp  (x)  =  e  <^  +  e       ^ . 

where  <:  is  a  constant,  or  any  function  which  takes  the  same 
value  when  x  is  changed  to  2  x/ 

The  solution  of  Laplace  and  D'Alembert,  applied  to 
the  statical  problem  of  the  preceding  section,  leads  to  the 
case  in  which  c-  is  a  function  of  x.  Further,  since  we  cannot 
admit  values  of  c  such  a.sa+i ù,  where  a,  i>  are  both  different 
from  zero,  we  have  three  possible  cases,  according  as  c  is 
real,  a  pure  imaginary,  or  infinite.^    Corresponding  to  these 


1  We  have  stated  above  (p.  53),  when  speaking  of  FoNCENEX' 
memoir,  that,  if  it  v?as  not  the  vv'ork  of  Lagrange,  it  was  certainly 
inspired  by  him.  This  opinion,  accepted  by  Genocchi  and  other 
geometers,  dates  from  Delambre.  The  distinguished  biographer 
of  Lagrange  puts  the  matter  in  the  following  words:  "//  (Za- 
gi-aiigé)  fournissait  à  Fonceiiex  la  parile  analyllque  de  ses  mémoires  en 
ltd  laissajtl  le  soin  de  développer  les  raisonnements  sur  lesqueh  portaiettl 
ses  formules.  En  effet,  on  remarque  drja  dans  ces  mémoires  (of 
Foncenex)  cede  marche  purement  analitique,  qui  depuis  a  fait  le 
caractère  des  grandes  productions  de  Lagrange.  II  avail  trouvè  tt?ie 
nouvelle  théorie  dii  levier".  Notices  sur  la  voie  et  les  ouvrages  de  M. 
le  Comic  Lagrange.  Mém.  Inst,  de  France,  classe  Math,  et  Physique, 
T.  Xm,  p.  XXXV  (1 8 1 2). 

2  Cf.  D'Alembert:  Sur  les  principes  de  la  Mécaniqtce  :  Mém.  de 
l'Ac.  des  Sciences  de  Paris  (1769).  —  Laplace:  Recherches  sur 
l'intrgraiion  des  equations  diffirentiellcs  :  Mém.  Ac.  sciences  de  Paris 
(savants  étrangers)  T.  VII  (1733).  Oeuvres  de  Laplace,  T.  Vili, 
p.   106 — 7. 

3  We  can  obtain  this  result  directly  by  integrating  the  equa- 


The  Three  Geometries, 


199 


three  cases,  we  have  three  possible  laws  for  the  composition 
of  forces,  and  consequently  three  distinct  types  of  equations 
connecting  the  sides  and  angles  of  a  triangle.  These  results 
are  brought  together  in  the  following  table,  where  k  denotes 
a  real  positive  number. 


Value  of  c 

Form  of  q)  (^) 

Trigonometri- 
cal equations 

Nature  of 
plane 

c  =  k 

X               X 

ek'^-e    T_2cosh^ 
k 

b        cos  p 

cosh-; . —  „ 

X'       sin  a 

hyperbolic 

c  =  ik 

i  X         ix 

,k  -\-e     k  =  2  cos  — 
'                                   k 

b       cos  p 

elliptic 

c  =  00 

X                       X 

e^+e       «'  =  2 

cos  6 

1  =-. — - 
sm  a 

parabolic 

Conclusion:  The  law  for  the  composition  of  forces  per- 
pendicular to  a  straight  line,  leads,  in  a  certain  sense,  to  the 
relations  which  hold  between  the  sides  and  angles  of  a 
triangle,  and  thus  to  the  geometrical  properties  of  the  plane 
and  of  space. 

This  fact  was  completely  established  by  A.  Genocchi 
[181 7 — 1889]  in  two  most  important  papers',  to  which  the 
reader  is  referred  for  full  historical  and  bibliographical 
notes  upon  this  question. 


tion  (2),  or,  what  amounts  to  the  same  thing,  equation  (l)  of 
S  6.  Cf.,  for  this,  the  elementary  method  employed  by  Cauchy 
for   finding   the  function  satisfying  (i).     Oeuvres  de  Cauchy ,  (sér.  2). 

T.  ni,  p.  106— 113. 

I  One  of  them  is  the  Memoir  referred  to  on  p.  19 1.  The 
other,  which  dates  from  1869,  is  entitled:  Dei  primi  principii  della 
meccanica  e  della  geometria  in  relazione  al  postulato  d'Euclide.  Annali 
della  Società  italiana  delle  Scienze  (3).     T.  II,  p.   153 — 189. 


Appendix  II. 

Clifford's  Parallels  and  Surface. 
Sketch  of  Clifford-Klein's  Problem. 


Clifford's  Parallels. 

§  I.  Euclid's  Parallels  are  straight  lines  possessing  the 
following  properties: 

a)  They  are  coplanar. 

b)  They  have  no  common  points. 

c)  They  are  equidistant. 

If  we  give  up  the  condition  (c)  and  adopt  the  views  of 
Gauss,  Lobatschewsky  and  Bolyai,  we  obtain  a  first  ex- 
tension of  the  notion  of  parallelism.  But  the  parallels  which 
correspond  to  it  have  very  few  properties  in  common  with 
the  ordinary  parallels.  This  is  due  to  the  fact  that  the  most 
beautiful  properties  we  meet  in  studying  the  latter  depend 
principally  on  the  condition  (c).  For  this  reason  we  are  led 
to  seek  such  an  extension  of  the  notion  of  parallelism,  that, 
so  far  as  possible,  the  new  parallels  shall  still  possess  the 
characteristics,  which,  in  Euclidean  geometry,  depend  on 
their  equidistance.  Thus,  following  W.  K.  Clifford  [1845 — 
1879],  we  give  up  the  property  of  coplanariiy,  in  the  definition 
of  parallels,  and  retain  the  other  two.  The  new  definition  of 
parallels  will  be  as  follows: 

Two  straight  lines,  iti  the  same  or  in  different  planes,  are 
called  parallel,  when  the  points  of  the  one  are  equidistant  from 
the  points  of  the  other. 


Clifford's  Parallels.  201 

§  2.    Two  cases,  then,  present  tlieip.selves,  according  as 
these  parallels  lie,  or  do  not  lie,  in  the  same  plane. 

The  case  in  which  the  equidistant  straight  lines  are 
coplanar  is  quickly  exhausted,  since  the  discussion  in  the 
earher  part  of  this  book  [§  8]  allows  us  to  state  that  the 
corresponding  space  is  the  ordinary  Euclidean.  We  shall, 
therefore,  suppose  that  the  two 
equidistant  straight  lines  r  and  s    T  


are  not  in  the  same  plane,  and 

that   the   perpendiculars    drawn 

from    r   to    J  are  equal.    Obvi-    s  , 

A  R 

ously  these  lines  will  also  be  per- 

,  ,  Fig.  71. 

pendicular  to  r.    Let  AA ,  BB 

be  two  such  perpendiculars  (Fig.  71).  The  skew  quad- 
rilateral ABB' A ,  which  is  thus  obtained,  has  its  four  angles 
and  two  opposite  sides  equal.  It  is  easy  to  see  that  the 
other  two  opposite  sides  AB,  AB'  are  equal,  and  that  the 
interior  alternate  angles,  which  each  diagonal — e.  g.  AB' — 
makes  with  the  two  parallels,  are  equal.  This  follows  from 
the  congruence  of  the  two  right-angled  triangles  AAB'  and 
ABB'. 

If  now  we  examine  the  solid  angle  at  A,  from  a  theorem 
valid  in  all  the  three  geometrical. systems,  we  can  write 

<C  AAB'  -f  <^  B'AB  >  -^  AAB  =  i  right  angle. 

This  inequality,  taken  along  with  the  fact  that  the  angles 
AB' A  and  B' AB  are  equal,  can  be  written  thus: 

<^  AAB'  4-  <^  AB' A  >  i  right  angle. 

Stated  in  this  way,  we  see  that  the  sum  of  the  acute 
angles  in  the  right-angled  triangle  AA B'  is  greater  than  a 
right  angle.  Thus  in  the  said  triangle  the  Hypothesis  of  the 
Obtuse  Angle  is  verified,  and  consequently  parallels  ?iot  iti  the 
same  plane  can  exist  only  in  the  space  of  Riemann. 


202 


Appendix  II.     Clifford's  Parallels  and  Surface. 


§  3.  Now  to  prove  that  in  the  elliptic  space  of  Riemann 
there  actually  do  exist  pairs  of  straight  lines,  not  in  the  same 
plane  and  equidistant,  let  us  consider  an  arbitrary  straight 
line  r  and  the  infinite  number  of  planes  perpendicular  to  it. 
These  planes  all  pass  through  another  line  r,  the  polar 
of  r  in  the  absolute  polarity  of  the  elliptic  space.  Any  line 
whatever,  joining  a  point  of  r  with  a  point  of/,  is  perpend- 
icular both  to  r  and  to  /,  and  has  a  constant  length,  equal 
to  half  the  length  of  a  straight  line.  From  this  it  follows 
that  r,  r  are  two  equidistant  straight  lines^  not  in  the  same 
plane. 

But  two  such  equidistants  represent  a  very  particular 
case,  since  all  the  points  of  r  have  the  same  distance  not 
only  from  /,  but  from  all  the  points  of  r. 


r 

r 

/ 

H 

A 

M 

/ 

W 

K 

B 

Fig.  72. 


To  establish  the  existence  of  straight  lines  in  which  the 
last  peculiarity  does  not  exist,  we  consider  again  two  lines 
r  and  /,  one  of  which  is  the  polar  of  the  other  (Fig.  72). 
Upon  these  let  the  equal  segments  AB^  AB'  be  taken,  each 
less  than  half  the  length  of  a  straight  line.  Joining  A  with 
A^  and  B  with  B' ,  we  obtain  two  straight  lines  ^,  b,  not 
polar  the  one  to  the  other,  and  both  perpendicular  to  the 
lines  r,  r . 

It  can  easily  be  proved  that  a,  b  are  equidistant.  To 
show  this,   take   a  segment  AH  upon  AA;   then  on  the 


The  Polars  as  Parallels.  203 

supplementary  line  ^  to  AHA,  take  the  segment  ^i^  equal  to 
AH.  If  the  poinfs  H  and  M  are  joined  respectively  with 
£^  and  B,  we  obtain  two  right-angled  triangles  A£H,  ABM, 
which,  in  consequence   of  our  construction,  are  congruent. 

We  thus  have  the  equality 

HB'  =  B3f. 

Now  if  H  and  B  are  joined,  and  the  two  triangles 
HBB'  and  HBM  zx^  compared,  we  see  immediately  that  they 
are  equal.  They  have  the  side  HB  common,  the  sides  HB' 
and  MB  equal,  by  the  preceding  result,  and  finally  BB'  and 
HM  are  also  equal,  each  being  half  of  a  straight  line. 

This  means,  in  other  words,  that  the  various  points  of 
the  straight  line  a  are  equidistant  from  the  line  b.  Now  since 
the  argument  can  be  repeated,  starting  from  the  line  b  and 
dropping  the  perpendiculars  to  a,  we  conclude  that  the  line 
HK^  in  addition  to  being  perpendicular  to  b,  is  also  perpend- 
icular to  a. 

We  remark,  further,  that  from  the  equality  of  the 
various  segments  AB,  HK,  A B\ . . .  the  equality  of  the  re- 
spective supplementary  segments  is  deduced,  so  that  the  two 
lines  a,  b,  can  be  regarded  as  equidistant  the  one  from  the 
other,  in  two  different  ways.  If  then  it  happened  that  the 
line  AB  were  equal  to  its  supplement,  we  would  have  the  ex 
ceptional  case,  which  we  noted  previously,  where  a,  b  are 
the  polars  of  each  other,  and  consequently  all  the  points  of 
a  are  equidistant  from  the  different  points  of  b. 

§  4.  The  non-planar  parallels  of  elliptic  space  were 
discovered  by  Clifford  in  1873.^  Their  most  remarkable 
properties  are  as  follows: 


1  The   two   different  segments,   determined  by  two  points  on 
a  straight  line,  are  called  supplementary. 

2  Preliminary  Sketch  of  Biquaternions.     Proc.  Lond.  Math.  Soc. 
Vol.  IV.  p.  381— 395(1873).  Clifford's  Mathematical  Papers,  p.  181—200. 


204 


Appendix  II.     Clifford's  Parallels  and  Surface. 


fi)  If  a  siraigJit  line  meets  two  parallels,  it  makes  with 
the»!  equal  eorrespo?iding  angles,  equal  interior  alternate 
angles,  etc. 

(ii)  If  in  a  skew  quadrilateral  the  opposite  sides  are 
equal  and  the  adjacent  angles  supplemcjitary,  then  the  opposite 
sides  are  parallel. 

Such  a  quadrilateral  can  therefore  be  called  a  ske:a 
parallelogram . 

The  first  of  these  two  theorems  can  be  immediately 
verified;  the  second  can  be  proved  by  a  similar  argument 
to  that  employed  in  §  3. 

(iii)  If  two  straight  lines  are  equal  and  parallel,  ajid 
their  extremities  are  suitably  joined,  we  obtain  a  skezv  paral- 
lelogram. 

This  result,  which  can   be  looked  upon,  in  a  certain 

sense,  as  the  converse  of  (ii),  can  also  be  readily  established. 

(iv)    Through  a?iy  point  (AI)  in  space,  which  does  not 

lie  on  the  polar  of  a  straight  line  (r),  two  parallels  can  be 

drawn  to  that  line. 

Indeed,  let  the  perpendicular  MN  be  drawn  from  M 
to  r,  and  let  N'  be  the  point  in  which  the  polar  of  MN 

meets  r  (Fig.  73).  From 
this  polar  cut  off  the  two 
segments  N'  M' ,  N'AI", 
equal  to  NM,  and  join  the 
points  M',  M"  to  M.  The 
two  lines  /,  r",  thus  ob- 
tained, are  the  required  par- 
allels. 

If  M  lay  on  the  polar  of  r,  then  MN  would  be 
equal  to  half  the  straight  line;  the  two  points  M' ,  M" 
would  coincide:  and  the  two  parallels  /,  r"  would  also 
coincide. 


Fig-  73- 


Properties  of  Clifford's  Parallels. 


205 


The  angle  between  the  t.vo  parallels  /,  r"  can  be 
measured  by  the  segment  MM",  which  the  two  arms  of  the 
angle  intercept  on  the  polar  of  its  vertex.  In  this  way  we 
can  say  that  half  of  the  angle  between  r  and  r",  that  is, 
the  angle  0/ parallelism,  is  equal  to  the  distance  of  parallelism. 

To  distinguish  the  two  parallels  /,  r",  let  us  consider  a 
helicoidal  movement  of  space,  with  MN  for  axis,  in  which 
the  pencil  of  planes  perpendicular  to  MJV,  and  the  axis  J/' J/  ' 
of  that  pencil,  obviously  remain  fixed.  Such  a  movement 
can  be  considered  as  the  resultant  of  a  translation  along  MJV, 
accompanied  by  a  rotation  about  the  same  axis:  or  by  two 
translations,  one  along  MN,  the  other  along  M'M".  If  the 
two  translations  are  of  equal  amount,  we  obtain  a  space 
vector. 

Vectors  can  be  right-handed  or  left-handed.  Thus,  referr- 
ing to  the  two  parallels  /,  r",  it  is  clear  that  one  of  them 
will  be  superposed  upon  r  by  a  right-handed  vector  of 
magnitude  AfJV,  while  the  other  will  be  superposed  on  r  by 
a  left-handed  vector  of  the  same  magnitude.  Of  the  two 
lines  r,  r",  one  could  be  called  the  right-handed  parallel 
and  the  other  the  left-handed  parallel  to  r. 

(v)  Two  right-handed  {or  left-handed)  parallels  to  a 
straight  line  are  I'ight-handed  {or  left-handed)  parallels  to 
each  other. 

Let  b,  c  be  two  right-hand- 
ed parallels  to  a.  From  the 
two  points  A,  A  of  a,  distant 
from  each  other  half  the  length 
of  a  straight  Hne,  draw  the 
perpendiculars  AB,  AB'  on  b, 
and  the  perpendiculars  AC, 
AC  on  c  (cf.  Fig.  74). 

The  lines  AB',  AC  are  the  polars  of  AB  and  AC. 

Therefore  ^  BAC  =  <^B'AC. 


B 

/^/ 

B" 

A 

A' 

Fig.  74- 

206  Appendix  II.     Clifford's  Parallels  and  Surface. 

Further^  by  the  properties  of  parallels 

AB  =  AB\  AC^AC. 

Therefore  the  triangles  ABC,  A JS C  are  equal 

Thus  it  follows  that 

BC  =  B'C. 

Again,  since 

BB'  =  AA  =  CC\ 
the  skew  quadrilateral  BBC' C  has  its  opposite  sides  equal. 

But  to  establish  the  parallelism  of  b,  c,  we  must  also 
prove  that  the  adjacent  angles  of  the  said  quadrilateral  are 
supplementary  (cf  ii).  For  this  we  compare  the  two  solid 
angles  B  {AB' C)  and  B'  (AB"C').  In  these  the  following 
relations  hold: 

^ABB'  =  -^  AB'B"  =  I  right  angle 
^  ABC  =  <^  AB'C. 

Further,  the  two  dihedral  angles,  which  have  BA  and 
B'A'  for  their  edges,  are  each  equal  to  a  right  angle,  dimin- 
ished (or  increased)  by  the  dihedral  angle  whose  normal 
section  is  the  angle  ABB'. 

Therefore  the  said  two  solid  angles  are  equal.  From 
this  the  equality  of  the  two  angles  B' BC,  B'B'C  follows. 
Hence  we  can  prove  that  the  angles  B,  B'  of  the  quadri- 
lateral BB'  C  C  are  supplementary,  and  then  (on  drawing 
the  diagonals  of  the  quadrilateral,  etc.)  that  the  angle  B  is 
supplementary  to  C,  and  C  supplementary  to  C,  etc. 

Thus  b  and  c  are  parallel.  From  the  figure  it  is  clear 
that  the  parallelism  between  b  and  c  is  right-handed,  if  that 
is  the  nature  of  the  parallelism  between  the  said  lines  and 
tlie  line  a. 

Clifford's  Surface. 

§  5.  From  the  preceding  argument  it  follows  that  all 
the  lifies  which  meet  three  right-handed  parallels  are  left-handed 
parallels  to  each  other. 


Clifford's  Surface. 


207 


Indeed,  if  ABC  is  a  transversal  cutting  the  three  lines 
a,  b,  c,  and  if  three  equal  segments  AA\  BB\  CC  are  taken 
on  these  lines  in  the  same  direction,"  the  points  A'B'C  lie 
on  a  line  parallel  to  ABC.  The  psjallelism  between  ABC 
and  A'B'C  is  thus  left-handed. 

From  this  we  deduce  that  three  parallels  a,  b,  c,  define 
a  ruled  surface  of  the  second  order  (Clifford's  Surface). 
On  this  surface  the  lines  cutting  a,  b,  c  form  one  system  of 
generators  {g^:  the  second  system  of  generators  {gd)  is 
formed  by  the  infinite  number  of  lines,  which,  like  a,  ^,  c, 
meet  {gs). 

Clifford's  Surface  possesses  the  following  charact- 
eristic properties: 

a)  Two  generators  of  the  same  system  are  parallel  to 
each  other. 

b)  Two  generators  of  opposite  systems  cut  each  other  at  a 
constant  atigle. 

§  6.  We  proceed  to  show  that  Clifford's  Surface  has 
t7vo  distinct  axes  of  rcvolutiofi. 

To  prove  this,  from 
any  point  M  draw  the 
parallels  d  (right-hand- 
ed), s  (left-handed),  to  a 
line  r,  and  denote  by  Ò 
the  distance  MN  of 
each  parallel  from  r 
(cf.  Fig.  75). 

Keeping  d  fixed,  let 
s  rotate  about  r,  and  let  /,  /',  /"  , 
positions  which  s  takes  in  this  rotation 


Fig.  75- 

.  be  the  successive 


I  It  is   clear  that   if  a   direction   is    fixed   for  one  line,    it  is 
then  fixed  for  every  line  parallel  to  the  first. 


208  Appendix  II.     Clifford's  Parallels  and  Surface. 

It  is  clear  that  s,  s',  s"  .  .  .  are  all  left-handed  parallels 
to  r  and  that  all  intersect  the  line  d. 

Thus  s  in  its  rotation  about  r  generates  a  Clifford's 
Surface. 

Vice  versa,  if  d  and  j-  are  two  generators  of  a  Clifford's 
Surface,  which  pass  through  a  point  M  of  the  surface,  and  2  Ò 
the  angle  between  them,  we  can  raise  the  perpendicular 
to  the  plane  sd  at  M  and  upon  it  cut  off  the  lines 
AIL  =  MiV  =  Ò. 

Let  Z>  and  ^  be  the  points  where  the  polar  of  ZiV  meets 
the  lines  d  and  s,  respectively,  and  let  i^be  the  middle  point 
ofZ'^=  2Ò. 

Then  the  lines  HL  and  HIV  are  parallel,  both  to  s 
and  d. 

Of  the  two  lines  HZ  and  HIV  choose  that  which  is 
a  right-handed  parallel  to  d  and  a  left-handed  parallel  to  s, 
say  the  line  HIV. 

Then  the  given  Clifford's  Surface  can  be  generated  by 
the  revolution  of  s  or  d  about  HIV. 

In  this  way  it  is  proved  that  every  Clifford's  Surface 
possesses  one  axis  of  rotation  and  that  every  point  on  the 
surface  is  equidistant  from  it. 

The  existence  of  another  axis  of  rotation  follows  im- 
mediately, if  we  remember  that  all  the  points  of  space,  equi- 
distant from  HN.,  are  also  equidistant  from  the  line  which  is 
the  polar  of  HN. 

This  line  will,  therefore,  be  the  second  axis  of  rotation 
of  the  Clifford's  Surface. 

§  7.  The  equidistance  of  the  points  of  Clifford's 
Surface  from  each  axis  of  rotation  leads  to  another  most 
remarkable  property  of  the  surfaces.  In  fact,  every  plane 
passing  through  an  axis  r  intersects  it  in  a  line  equidistant 
from  the  a.xis.  The  points  of  this  line,  being  also  equally 
distant  from  the  point  {O)  in  which  the  plane  of  section  meets 


The  Axes  of  Clifford's  Surface. 


209 


the  other  axis  of  the  surface,  lie  on  a  circle,  whose  centre  (O) 
is  the  pole  of  /■  with  respect  to  the  said  line.  Therefore  the 
meridians  and  the  parallels  of  the  surface  are  circles. 

The  surface  can  thus  be  generated  by  making  a  circle 
rotate  about  the  polar  of  its  cetitre,  or  by  making  a  circle  move 
so  that  its  centre  describes  a  straight  line,  while  its  plane  is 
maintained  constantly  perpendicular  to  it  (Bianchi).' 

This  last  method  of  generating  the  surface,  common 
also  to  the  Euclidean  cylinder,  brings  out  the  analogy  be- 
tween Clifford's  Surface  and  the  ordinary  circular  cyhnder 
This  analogy  could  be  carried  further,  by  considering  the 
properties  of  the  hehcoidal  paths  of  the  points  of  the  surface, 
when  the  space  is  submitted  to  a  screwing  motion  about 
either  of  the  axes  of  the  surface. 

§  8.  Finally,  we  shall  show  that  the  geometry  on  Clif- 
ford's Surface,  understood  in  the  sense  explained  in  §§  67, 
68,  is  identical  with  Euclidean  geometry. 

To  prove  this,  let  us  determine  the  law  according  to 
which  the  element  of  distance  between  two  points  on  the 
surface  is  measured. 

Let  u,  V,  be  respectively  a  parallel  and  a  meridian 
through  a  point  O  on  the  surface,  and  M  any  arbitrary  point 
upon  it. 

Let  the  meridian  and  parallel 
through  M  cut  off  the  arcs  OP,  OQ 
from  u  and  v.  The  lengths  u,  ?>  of 
these  arcs  will  be  the  coordinates  of  Q 
Jlf.  The  analogy  between  the  system 
of  coordinates  here  adopted  and  the 
Cartesian  orthogonal  system  is  evident  0 
(cf.  Fig.  76).  Fig.  75. 


I  Sulla  siipeificie  a  curvatiaa  nulla  in  geometria  ellittica.  Ann. 
di  Mat.  (2)  XXIV,  p.  107  (1896).  Also  Lezioni  di  Geometria  Differ- 
enziale.    2a  Ed.,  Voi.  I,  p.  454  (Pisa,  1902). 

14 


2 IO  Appendix  II.     Clifford's  Parallels  and  Surface. 

Let  M'  be  a  point  whose  distance  from  M  is  infini- 
tesimal. If  {u,  v)  are  the  coordinates  of  J/,  we  can  take 
{u  +  du,  V  +  dv)  for  those  of  M' . 

Now  consider  the  infinitesimal  triangle  MM' N.,  whose 
third  vertex  N  is  the  point  in  which  the  parallel  through  AI 
intersects  the  meridian  through  M' .  It  is  clear  that  the  angle 
MNM'  is  a  right  angle,  and  that  the  sides  MN,  NM'  are 
equal  to  du^  dv. 

On  the  other  hand,  this  triangle  can  be  regarded  as 
rectilinear  (as  it  lies  on  the  tangent  plane  at  M).  So  that, 
from  the  properties  of  infinitesimal  plane  triangles,  its  hypo- 
tenuse and  its  sides,  by  the  Theorem  of  Pythagoras,  are  con- 
nected by  the  relation 

ds^  =  du^  -^  dv^. 
But  this  expression  for  ds*  is  characteristic  of  ordinary 
geometry,   so  that  we  can  immediately  deduce  that  the  pro- 
perties of  the  Euclidean  plane  hold  i?i  every  normal  region  on 
a  Clifford's  Surface. 

An  important  application  of  this  result  leads  to  the 
evaluation  of  the  area  of  this  surface.  Indeed,  if  we  break 
it  up  into  such  congruent  infinitesimal  parallelograms  by 
means  of  its  generators,  the  area  of  one  of  these  will  be 
given  by  the  ordinary  expression 

dx  dy  sin  9, 
where  dx,  dy  are  the  lengths  of  the  sides  and  0  is  the  con- 
stant angle  between  them  (the  angle  between  two  generators). 
The  area  of  the  surface  is  therefore 

E  dx  dy  sin  0  =  sin  9  2  dx  •  2  dy. 

But  both  the  sums  2  dx,  2  dy  represent  the  length  /  of 
a  straight  line. 

Therefore  the  area  A  of  Clifford's  Surface  takes  the 
very  simple  form. 


The  Area  of  Clifford's  Surface.  211 

A  =  /^  sin  e, 
which  is  identical  v/ith  the   expression  for  the   area  of  a 
EucHdean  parallelogram  (Clifford).' 

Sketch  of  Clifford-Klein's  Problem. 

§  9.  Clifford's  ideas,  explained  in  the  preceding 
sections,  led  Klein  to  a  new  statement  of  the  fundamental 
problem  of  geometry. 

In  giving  a  short  sketch  of  Klein's  views,  let  us  refer 
to  the  results  of  §  68  regarding  the  possibility  of  interpret- 
ing plane  geometry  by  that  on  the  surfaces  of  constant 
curvature.  The  contrast  between  the  properties  of  the  Eu- 
chdean  and  Non-Euclidean  planes  and  those  of  the  said 
surfaces  was  there  restricted  to  suitably  bounded  regions. 
In  extending  the  comparison  to  the  unbounded  regions,  we 
are  met,  in  general,  by  differences;  in  some  cases  due  to 
the  presence  of  singular  points  on  the  surfaces  (e.  g.,  vertex 
of  a  cone);  in  others,  to  the  different  connectivities  of  the 
surfaces. 

Leaving  aside  the  singular  points,  let  us  take  the  cir- 
cular cylinder  as  an  example  of  a  surface  of  constant  curv- 
ature, everywhere  regular,  but  possessed  of  a  connectivity 
different  from  that  of  the  Euclidean  plane. 

The  difference  between  the  geometry  of  the  plane  and 
that  of  the  cylinder,  both  understood  in  the  complete  sense, 
has  been  already  noticed  on  p.  140,  where  it  was  observed 
that  the  postulate  of  congruence  between  two  arbitrary 
straight  lines  ceases  to  be  true  on  the  cylinder.  Nevertheless 
there  are  numerous  properties  common  to  the  two  geometries, 


I  Preliminary  Sketch,  cf.  p.  203  above.  The  properties  of 
this  surface  were  referred  to  only  very  briefly  by  Clifford  in  1873. 
They  are  developed  more  fully  by  Klein  in  his  memoir:  Zur  nichl- 
euklidischen  Geometrie,  Math.  Ann.  Bd.  XXXVII,  p.  544—572  (1890). 

14* 


212  Appendix  II.     Clifford's  Parallels  and  Surface. 

which  have  their  origin  in  the  double  characteristic,  that 
both  the  plane  and  the  cylinder  have  the  same  curvature, 
and  that  they  are  both  regular. 

These  properties  can  be  summarized  thus: 

i)  The  geometry  of  a?iy  normal  region  of  the  cylinder 
is  identical  with  that  of  any  normal  region  of  the  plane. 

2)  The  geometry  of  any  normal  region  whatsoever  of 
the  cylinder,  fixed  with  respect  to  an  arbitrary  point  upon  it, 
is  identical  with  the  geometry  of  any  normal  region  what- 
soever of  the  plane. 

The  importance  of  the  comparison  between  the  ge- 
ometry of  the  plane  and  that  of  a  surface,  founded  on  the 
properties  (i)  and  (2),  arises  from  the  following  consid- 
erations : 

A  geometry  of  the  plane,  based  upon  experimental 
criteria,  depends  on  two  distinct  groups  of  hypotheses.  The 
first  group  expresses  the  validity  of  certain  facts,  directly 
observed  in  a  region  accessible  to  experiment  {postulates  of 
the  normal  region);  the  second  group  extends  to  inaccessible 
regions  some  properties  of  the  initial  region  {postulates  of 
extension). 

The  postulates  of  extension  could  demand,  e.  g.,  that 
the  properties  of  the  accessible  region  should  be  valid  in  the 
entire  plane.  We  would  then  be  brought  to  the  two  forms, 
the  parabolic  and  the  hyperbolic  plane.  If,  on  the  other  hand, 
the  said  postulates  demanded  the  extension  of  these  pro- 
perties, with  the  exception  of  that  which  attributes  to  the 
straight  line  the  character  of  an  open  line,  we  ought  to  take 
account  of  the  elliptic  plane  as  well  as  the  two  planes  mentioned. 

But  the  preceding  discussion  on  the  regular  surfaces  of 
constant  curvature  suggests  a  more  general  method  of  enun- 
ciating the  postulates  of  extension.  We  might,  indeed,  simply 
demand  that  the  properties  of  the  initial  region  should  hold 
in  the  neighbourhood  of  every  point  of  the  plane.    In  this 


Clifford-Klein's  Problem.  21 3 

case,  the  class  of  possible  forms  of  planes  receives  con- 
siderable additions.  We  could,  e.  g.,  conceive  a  form  with 
zero  curvature,  of  double  connectivity,  and  able  to  be  com- 
pletely represented  on  the  cyhnder  of  Euclidean  space. 

The  object  of  Clifford-Klein' s  problem  is  the  determination 
of  all  the  two  dimensional  manifolds  of  constant  curvature, 
which  are  everyiohere  regular. 

§  10.  Is  it  possible  to  realise,  with  suitable  regular 
surfaces  of  constant  curvature,  in  the  Euclidean  space,  all 
the  for7tis  of  Clifford-Klein  ? 

The  answer  is  in  the  negative,  as  the  following  example 
clearly  shows.  The  only  regular  developable  surface  of  the 
Euclidean  space,  whose  geometry  is  not  identical  with  that 
of  the  plane,  is  the  cylinder  with  closed  cross-section.  On 
the  other  hand,  Clifford's  Surface  in  the  elliptic  space  is  a 
regular  surface  of  zero  curvature,  which  is  essentially  different 
from  the  plane  and  cylinder. 

However  with  suitable  conventions  we  can  represent 
Clifford's  Surface  even  in  ordinary  space. 

Let  us  return  again  to  the  cylinder.  If  we  wish  to  un- 
fold the  cylinder,  we  must  first  render  it  simply  connected 
by  a  cut  along  a  generator  {g);  then,  by  bending  without 
stretching,  it  can  be  spread  out  on  the  plane,  covering  a 
strip  between  two  parallels  igxigz)- 

There  is  a  one-one  correspondence  between  the  points 
of  the  cylinder  and  those  of  the  strip.  The  only  exception  is 
afforded  by  the  points  of  the  generator  (^),  to  each  of  which 
correspond  two  points,  situated  the  one  on^i,  the  other  on 
g2.  However,  if  it  is  agreed  to  regard  these  two  points  as 
idefitical,  that  is,  as  a  single  point,  then  the  correspondence 
becomes  one-one  without  exception,  and  the  geometry  of  the 
strip  is  completely  identical  with  thai  of  the  cylinder. 


214  Appendix  II.     Clifford's  Parallels  and  Surface. 

A  representation  analogous  to  the  above  can  also  be 
adopted  for  Clifford's  Surface.  First  the  surface  is  made 
simply  connected  by  two  cuts  along  the  intersecting  gener- 
ators {g,  g).  In  this  way  a  skew  parallelogram  is  obtained 
in  the  elliptic  space.  Its  sides  have  each  the  length  of  a 
straight  line,  and  its  angles  G  and  9'  [O  +  0'=  2  right  angles] 
are  the  angles  between  g  and  g. 

This  being  done,  we  take  a  rhombus  in  the  Eu- 
clidean plane,  whose  sides  are  the  length  of  the  straight  line 
in  the  elliptic  plane,  and  whose  angles  are  0,  6'.  On  this 
rhombus  Clifford's  Surface  can  be  represented  congruenti}' 
(developed).  The  correspondence  between  the  points  of  the 
surface  and  those  of  the  rhombus  is  a  one-one  correspond- 
ence, with  the  exception  of  the  points  of^  and^',  to  each 
of  which  correspond  two  points,  situated  on  the  opposite 
sides  of  the  rhombus.  However,  if  we  agree  to  regard  these 
points  as  identical,  two  by  two,  then  the  correspondence 
becomes  one-one  without  exception,  and  the  geometry  of 
the  rhombus  is  completely  identical  7oith  that  of  Clifford's 
Surface.'^ 

§  II.  These  representations  of  the  cylinder  and  of 
Clifford's  Surface  show  us  how,  for  the  case  of  zero  curva- 
ture, the  investigation  of  Clifford-Klein's  forms  can  be 
reduced  to  the  determination  of  suitable  Euclidean  polygons, 
eventually  degenerating  into  strips,  whose  sides  are  two  by 
two  transformable,  one  into  the  other,  by  suitable  movements 
of  the  plane,  their  angles  being  together  equal  to  four  right- 
angles  (Klein).*  Then  it  is  only  necessary  to  regard  the 
points  of  these  sides  as  identical,  two  by  two,  to  have  a 
representation  of  the  required  forms  on  the  ordinary  plane. 

I  Cf.  Clifford  loc.  cit.  Also  Klei.n's  memoir  referred  to 
on  p.  2X1. 

*  Cf.  the  memoir  just  named. 


Clifford-Klein's  Problem.  215 

It  is  possible  to  present,  in  a  similar  way,  the  investi- 
gation of  Clifford-Klein's  forms  for  positive  or  negative 
values  of  the  curvature,  and  the  extension  of  this  problem 
to  space.' 


I  A  systematic  treatment  of  Clifford-Klein's  problem  is  to 
be  found  in  Killing's  Eiiifilhrung  in  die  Gnindlagen  der  Geometrie. 
Bd.  I,  p.  271 — 349  (Paderborn,  1893). 


Appendix  III. 

The  Non-Euclidean  Parallel  Construction 
and  other  Allied  Constructions. 

§  I.  The  Non-Euclidean  Parallel  Construction  depends 
upon  the  correspondence  between  the  right-angled  triangle 
and  the  quadrilateral  with  three  right  angles.  Indeed,  when 
this  correspondence  is  known,  a  number  of  different  con- 
structions are  immediately  at  our  disposal.* 

To  express  this  correspondence  we  introduce  the 
following  notation: 

In  the  right-angled  triangle,  as  usual,  a,  b  are  the  sides: 
c  is  the  hypotenuse:  X  is  the  angle  opposite  a  and  fi 
that  opposite  b.  Further  the  angles  of  parallelism  for  a,  b 
are  denoted  by  a  and  p:  and  the  lines  which  have  X,  ]x.  for 
angles  of  parallelism  are  denoted  by  /,  tn.  Also  two  lines, 
for  which  the  corresponding  angles  of  parallelism  are  com- 
plementary, are  distinguished  by  accents,  e.  g.: 

n  {d)  =  I  -  n(^),  n(/')  =  ^  -  ^  (^^- 

Then  with  this  notation:  To  every  right-angled  triangle 
{a,  b,  c,  X,  \x)  there  corresponds  a  quadrilateral  with  three 
right-angles^  whose  fourth  angle  (acute)  is  P,  a7id  whose  sides 
are  c,  m\  a,  /,  taken  in  order  from  the  corner  at  which  the 
angle  is  p. 

The  converse  of  this  theorem  is  also  true. 


I  Cf.  p.  256  of  Engel's  work  referred  to  on  p. 


Correspondence  between  Quadrilateral  and  Triangle.      217 

The  following  is  one  of  the  constructions,  which  can  be 
derived  from  this  theorem,  for  drawing  the  parallel  through 
A  to  the  line  BC  (cf.  Fig.  77). 

Let  AB  be  the  perpendicular  from  A  to  BC.  At  A  draw 
the  line  perpendicular  to  AB,  and  from  any  point  C  in  BC 
draw  the  perpendicular  CD     ^  3 

to  this  line. 

With  centre  A  and  rad- 
ius BC  (equal  to  c)  describe 
a  circle  cutting  CD  in  E. 

Now  we  have 

^  EAD  =  M, 
and  therefore 

•^  BAE  =  -^  —  ^  =  n  (;//). 

But  the  sides  of  the  quadrilateral  are  c,  m',  a,  /,  taken  in 
order  from  C. 

Therefore  A£  is  parallel  to  BC. 

If  a  proof  of  this  construction  is  required  without  using 
the  trigonometrical  forms,  one  might  attempt  to  show  direct- 
ly that  the  line  AE  produced,  (simply  owing  to  the  equality 
of  BC  and  A£),  does  not  cut  BC  produced,  and  that  the 
two  have  not  a  common  perpendicular.  If  this  were  the 
case,  they  would  be  parallel.  Such  a  proof  has  not  yet  been 
found. 

Again,  we  might  prove  the  truth  of  the  construction 
using  the  theorem,  that  in  a  prism  of  triangular  section  the 
sum  of  the  three  dihedral  angles  is  equal  to  two  right  angles': 
so  that  for  a  prism  with  n  angles  the  sum  is  (2  n—4)  right 
angles.    This  proof  is  given  in  §  2  below. 

.  ^  Cf.  LoBATSCHEWSKY  (Engel's  translation)  p.  172. 


2l8        Appendix  III.    The  Non-Euclidean  Parallel  Construction. 


Finally,  the  correspondence  stated  in  the  above  theorem 
— only  part  of  which  is  required  for  the  Parallel  Construction 
of  Fig.  78  ■ —  can  be  verified  without  the  use  of  the  geo- 
metry of  the  Non-Euclidean  space.  This  proof  is  given  in  S  3- 

§  2.  Direct  proof  of  the  Parallel  Construction  by  fneans 
of  a  Prism. 

Q 


Fig.  78. 


Let  ABCD  be  a  plane  quadrilateral  in  which  the  angles 
at  Z>,  Ay  B  are  right  angles.  Let  the  angle  at  C  be  denoted 
by  p,  AD  by  a,  DC  by  /,  CB  by  c,  and  BA  by  m. 

At  A  draw  the  perpendicular  ^Q  to  the  plane  of  the 
quadrilateral.  Through  B,  C,  and  Z?  draw  ^Q,  CQ  and  Z>S2 
parallel  to  A^. 

Also  through  A  draw  AQ  parallel  to  BC,  cutting  CD 
in  E  {ED  =  b^,  and  let  the  plane  through  A^  and  AE 
cut  CZPQ  in  EQ..    From  the  definition,  we  have 


^EAD 


n  (?//) 


Further  the  plane  ^lAB  is  at  right  angles  to  a,  and  the 
plane  Q.DA  at  right  angles  to  /,  since  ^A  and  AB  are  per- 
pendicular to  a,  while  QZ>  and  a;  are  perpendicular  to  /. 


Direct  Proof  of  the  Parallel  Construction.  2I9 


IT 


Also        <^  AB9.  =  <^  OAB  =  -^  —  ^ 


In  the  prism  Q  {ABCD)  the  faces  which  meet  in  Q^, 
^.B,  QD  are  perpendicular.  Also  the  four  dihedral  angles 
make  up  four  right  angles.  It  follows  that  the  faces  of  the 
prism  C  (DBQ),  which  meet  along  CQ,  are  perpendicular. 
Also  it  is  clear  that  in  £  (DQA)  the  faces  which  meet  in  £A 
are  perpendicular,  while  the  dihedral  angle  for  the  edge  CD 
is  the  same  as  for  £D  (thus  equal  to  a). 

We  shall  now  prove  the  equality  of  the  other  dihedral 
angles  in  these  prisms  C  {DBQ.)  and  E  (DQA) — those  con- 
tained by  the  faces  which  meet  in  CB  and  AE. 

In  the  first  prism  this  angle  is  equal  to  the  angle  be- 
tween the  planes  ABCD  and  CBQ.    It  is  thus  equal  to 


|U,  i.  e.  it  is  equal  to  <^  ABQ.. 


In  the  second  prism,  the  angle  between  the  planes 
meeting  in  EQ  belongs  also  to  the  prism  Q  {ADE).  In  this 
the  angle  at  Q.D  is  a  right-angle,  and  that  at  QA  is  equal 

IT 

to  H-    Thus  the  third  angle  is  equal  to |li. 

Therefore  the  prisms  C  {DB9.)  and  E  (DQJ)  are 
congruent 

Therefore         ^  BCQ  =  ^  QEA, 
and  the  lines  which  have  these  angles  of  parallelism  are 
also  equal. 

Thus  c  =  BC  and  ^i  =  AE 

are  equal,  which  was  to  be  proved. 
Further  it  follows  that 

^  DEA  =  <^  DCQ; 
i.  e.the  angle  Xj,  opposite  the  side  a  of  the  triangle,  is  given  by 

X^  =  TT  (/)  =  X. 
Finally  ^  DCB  ==  ^  DEQ; 

i.e.  P  =  17  (d,),  or  Ù,  =  a. 


220       Appendix  III.    The  Non-Euclidean  Parallel  Construction. 

Thus  the  correspondence  between  the  triangle  and  the 
quadrilateral  is  proved.^ 

§  3.    Proof  of  the  Correspondence  by  Plane  Geometry. 

In  the  right-angled  triangle  ABC  produce  the  hypo- 
tenuse AB  to  D,  where  the  perpendicular  at  D  is  parallel  to 
C^(cf.  Fig.  79). 


Fig-  79- 


Then  with  the  above  notation 

BD  =  m. 

Draw  through  A  the  parallel  to  Z>0  and  CBQ. 

Then 

^  CAQ  =  p  =  n  {b), 

and  it  is  also  equal  to 

X  +  <C  DA(ò  =  \  +  TT  (^  +  w). 

We  thus  obtain  the  first  of  the  six  following  equations.^ 

The  third  and  fifth  can  be  obtained  in  the  same  way.    The 

second,  fourth,  and  sixth,  come  each  from  the  preceding,  if 

we  interchange  the  two  sides  a  and  b^  and,  correspondingly 

the  angles  X  and  )li. 

1  Bonola:  1st.  Lombardo,  Rend.  (2).  T.  XXXVII,  p.  255 — 
258  (1904).  The  theorem  had  already  been  proved  by  pure 
geometrical  methods  by  F.  Engel:  Bull,  de  la  Soc.  Phys.  Math. 
de  Kasan  (2).  T.  VI  (1896);  and  Bericht  d.  Kon.  Sachs.  Ges.  d. 
Wiss.,  Math.-Phys.  Klasse,  Bd.  L,  p.  181—187  (Leipzig,  1898). 

2  Cf.  LoBATSCHEWSKY  (Engel's  translation),  p.  15 — 16,  and 
LlEBMANN,  Math.  Ann.  Bd.  LXI,  p.   1S5,  (1905). 


Second  Proof  of  the  Parallel  Construction.  221 

The  table  for  this  case  is  as  follows: 
\  +  TT  (^  +  w)  =  p,     ^  +  U  (c  +  /)  =  a: 

\  +  p  =  TT  (f  —  w),     ^  +  a  =  TT  (^  —  /); 

T](è+/)+  Uim  —  a)^^jx,  U (m  +  a)  +  U  (I—  à)  =^-rx. 

Similar  equations  can  also  be  obtained  for  the  quad- 
rilateral with  three  right  angles.  Some  of  the  sides  have  to 
be  produced^  and  the  perpendiculars  drawn^  which  are 
parallel  to  certain  other  sides,  etc. 

If  we  denote  the  acute  angle  of  the  quadrilateral  by  p,, 
and  the  sides,  counting  from  it,  by  c^,  m/,  a^,  and  /i,  we  ob- 
tain the  following  table: 

K  +  Tl  {c;  +  m,)  =  p, ,      T.  +  n  (/,  -h  a,')  =  P,; 

K  +  px  =  n  (c,  —m,),      Tx  +  Pi  =  n  (/,  —  a^'); 

The  second,  fourth,  and  sixth  formulae  come  from  inter- 
changing Ci  and  ;«i',  with  /i  and  Ui ,  as  in  the  right-angled 
triangle. 

Let  us  now  imagine  a  right-angled  triangle  constructed 
with  the  hypotenuse  c  and  the  adjacent  angle  \x\  and  let  the 
remaining  elements  be  denoted  by  a,  b,  X  as  above. 

In  the  same  way,  let  a  quadrilateral  with  three  right- 
angles  be  constructed,  in  which  c  is  next  the  acute  angle,  m 
follows  c,  the  remaining  elements  being  a^,  /, ,  and  p,. 

Then  a  comparison  of  the  first  and  third  formulae  for 
the  triangle,  with  the  first  and  third  for  the  quadrilateral, 
shows  that 

Pi  =  Pj   ^i  =  ^• 

The  fifth  formula  of  both  tables  then  gives 

Ui  =  a. 
Hence  the  theorem  is  proved. 


222       Appendix  III.    The  Non-Euclidean  Parallel  Construction. 


From  the  two  tables  it  also  follows  that  to  a  right- 
angled  triangle  with  the  elements 

a,  b,  c,  X,  \x, 
there  corresponds  a  second  triangle  with  the  elements 

IT 

a,  =  a,  b^<==  I ,   c^  =  m,  \i  =  — P,  \x^  =  ^ , 

a  result  which  is  of  considerable  importance  in  further  con- 
structions.   But  we  shall  not  enter  into  fuller  details. 

The  possibility  of  the  Non-Euclidean  Parallel  Construc- 
tion, with  the  aid  of  the  ruler  and  compass,  allows  us  to 
draw,  with  the  same  instruments,  the  common  perpendicular 
to  two  lines  which  are  not  parallel  and  do  not  meet  each 
other  (the  non-intersecting  lines);  the  common  parallel  to  the 
two  Hues  which  bound  an  angle;  and  the  line  which  is  per- 
pendicular to  one  of  the  bounding  lines  of  an  acute  angle 
and  parallel  to  the  other.  We  shall  now  describe,  in  a  few 
words,  how  these  constructions  can  be  carried  out,  following 
the  lines  laid  down  by  Hilbert.^ 

§  4.  Construction  of  the  common  perpendicular  to  two 
non-intersecting  straight  lines. 


Fig.  80. 

Let  a  =  Ai_A^  b  =  Bj,B,  be  two  non-intersecting  lines; 
that  is,  lines  which  do  not  meet  each  other,  and  are  not 
parallel  (cf.  Fig.  80). 


I  Neue  Begiiindung  der  Bolyai- Lobatschefskyschen  Geometrie. 
Math.  Ann.  Bd.  57,  p.  137 — 150  (1903).  Hilbert's  Gruyidlagen  der 
Geometrie,  2.  Aufl.,  p.   ro7  at  seq. 


Some  Allied  Constructions.  223 

Let  AiB,,  AB  be  the  perpendiculars  drawn  from  the 
points  Al ,  A  upon  a  to  the  Hne  b,  constructed  as  in  ordinary 
geometry. 

If  the  segments  A^Bi,  AB,  are  equal,  the  perpendicular 
to  b  from  the  middle  point  of  the  segment  B^B  is  also  per- 
pendicular to  a;  so  that,  in  this  case,  the  construction  of 
the  common  perpendicular  is  already  effected. 

If,  on  the  other  hand,  the  two  segments  AiB^,  AB  are 
unequal,  let  us  suppose,  e.  g.,  that  A^Bi  is  greater  than  AB. 

Then  cut  off  from  A^Bi  the  segment  A'Bj,  equal  to  AB; 
and  through  the  point  A',  in  the  part  of  the  plane  in  which 
the  segment  AB  lies,  let  the  ray  A'M'  be  drawn,  such  that 
the  angle  B^A'iM'  is  equal  to  the  angle  which  the  line  a 
makes  with  AB  (cf.  Fig.  80). 

The  ray  A'M'  must  cut  the  line  a  in  a  point  M'  (cf. 
Hilbert,  loc.  cit.).  From  M'  drop  the  perpendicular  M' P' 
to  b^  and  from  the  line  a,  in  the  direction  A-^A^  cut  off  the 
segment  ^^  equal  to  AM'. 

If  the  perpendicular  MP  is  now  drawn  to  b,  we  have  a 
quadrilateral  ABPM  which  is  congruent  with  the  quad- 
rilateral A'B^P'M'. 

It  follows  that  MF  is  equal  to  M'F'. 

It  remains  only  to  draw  the  perpendicular  to  b  from 
the  middle  point  of  P'F  to  obtain  the  common  perpendicular 
to  the  two  lines  a  and  b. 

§  5.  Construction  of  the  common  parallel  to  two  straight 
lines  which  bound  any  angle. 

Let  a  =  AO,  and  b  =  BO,  be  the  two  lines  which  con- 
tain the  angle  A  OB  (cf.  Fig,  81).  From  a  and  b  cut  off  the 
equal  segments  OA  and  OB;  and  draw  through  A  the  ray 
b'  parallel  to  the  line  b,  and  through  B  the  ray  a'  parallel  to 
the  line  a. 


224       Appendix  III.    The  Non-Euclidean  Parallel  Construction. 


Let  «I  and  ^i  be  the  bisectors  of  the  angles  contained 
by  the  lines  ab\  and  db. 

The  two  lines  a^b^  are  non-intersecting  lines,  and  their 
common  perpendicular  yii^i,  the  construction  for  which  was 
given  in  the  preceding  paragraph,  is  the  common  parallel  to 
the  lines  which  bound  the  angle  AOB. 


\B' 


A,  B, 

Fig.  8i. 

Reference  should  be  made  to  Hilbert's  memoir,  quot- 
ed above,  for  the  proof  of  this  construction. 

§  6.  Construction  of  the  straight  line  7vhich  is  perpendi- 
cular to  one  of  the  lines  bounding  ati  acute  angle  and  parallel 
to  the  other. 

Let  a  =  AO  and  b  =  BO,  be 

the  two  lines  which  contain  the  acute 

angle  ^C>j9;  and  let  the  ray  b'  =  B' O 

be  drawn,  the  image  of  the  line  b  in 

0  a  (cf.  Fig.  82). 

Then,  using  the  preceding  con- 
struction^ let  the  line  BiBj,'  be  drawn 
parallel  to  the  two  lines  which  con- 
tain  the  angle  BOB . 

This  line,  from  the  symmetry  of 
the  figure  with  respect  to  a,  is  perpendicular  to  OA. 

It  follows  that  BiB\  is  parallel  to  one  of  the  lines  which 
contain  the  angle  AOB  and  perpendicular  to  the  other. 

^.?  7.  The  constructions  given  above  depend  upon 
metrical  considerations.  However  it  is  also  possible  to  make 
use  of  the  fact  that  to  the  metrical  definitions  of  perpend- 


B' 


Fig.  Sz. 


Projective  Constructions.  225 

icularity  and  parallelism  a  projective  meaning  can  be  given 
(§  79),  and  that  projective  geometry  is  independent  of  the 
parallel  postulate  (§  80). 

Working  on  these  lines,  what  will  be  the  construction 
for  the  parallels  through  a  point  A  to  a.  given  line? 

Let  the  points  /'i,  1*2,  P^  ^^^  P^i  ^2',  P^  be  given 
on  g  so  that  the  points  P^ ,  P^  •,  P^,  are  all  on  the  same 
side  of  Pi,  Pi,  P^,  and 

p,p,'  =  p,p;  =  p,p;. 

Join  AP-i,  ÀP2,  APt^  and  denote  these  Hnes  by  s^,  s^, 
and  Sy  Similarly  let  AP^',  AP^',  AP.'  be  denoted  by  Si', 
$2  and  J3'.  Then  the  three  pairs  of  rays  through  A^  determ- 
ine a  projective  transformation  of  the  pencil  is)  into  itself, 
the  double  elements  of  which  are  obviously  the  two  parallels 
which  we  require.  These  double  elements  can  be  constructed 
by  the  methods  of  projective  geometry.^ 

The  absolute  is  then  determined  by  five  points:  i.  e.,  by 
five  pairs  of  parallels;  and  so  all  further  problems  of  metrical 
geometry  are  reduced  to  those  of  projective  geometry. 

If  we  represent  (cf.  §  84)  the  Lobatschewsky-Bolyai 
Geometry  (e.  g.,  for  the  Euclidean  plane)  so  that  the  image 
of  the  absolute  is  a  given  conic  (not  reaching  infinity),  then 
it  has  been  shown  by  Grossmann^  that  most  of  the  problems 
for  the  Non-Euchdean  plane  can  be  very  beautifully  and 
easily  solved  by  this  'translation'.  However  we  must  not 
forget  that  this  simplicity  disappears,  if  we  would  pass  from 
the  'translation'  back  to  the  'original  text'. 


1  Cf.  for  example,  Enriques,  Geometria  proiettiva,  (referred  to 
on  p.  156)  S  73- 

2  Gross.mann,  Die  fiiiidamentalen  Konstriiklioneti  der  nicht- 
eiiklidiscken  Geometrie,  Programm  der  Thurgauischen  Kantonschule, 
(Frauenfeld,   1904). 

15 


226     Appendix  III.     The  Non-Euclidean  Parallel  Construction. 

In  the  Non-Euclidean  plane  the  absolute  is  inaccessible, 
and  its  points  are  only  given  by  the  intersection  of  pencils 
of  parallels.  The  points  Outside  of  the  absolute,  while  they 
are  accessible  in  the  'translation',  cannot  be  reached  in  the 
'text'  itself.  In  this  case  they  are  pencils  Of  straight  lines, 
which  do  not  meet  in  a  point,  but  go  through  the  (ideal) 
pole  of  a  certain  line  with  respect  to  the  absolute. 

If,  then,  we  would  actually  carry  out  the  constructions, 
difficulties  will  often  arise,  such  as  those  we  meet  in  the 
translation  of  a  foreign  language,  when  we  must  often  sub- 
stitute for  a  single  adjective  a  phrase  of  some  length. 


Appendix  IV. 

The  Independence  of  Projective  Geometry 
from  Euclid's  Postulate. 

§  I.  Statement  of  the  Frobietn.  In  the  following  pages 
we  shall  examine  more  carefully  a  question  to  which  only 
passing  reference  was  made  in  the  text  (cf.  §  80),  namely,  the 
validity  ofProjective  Geometry  in  Non-Euclidean  Space,  since 
this  question  is  closely  related  to  the  demonstration  of  the 
independence  of  that  geometry  from  the  Fifth  Postulate. 

In  elliptic  space  (cf  §  80)  we  may  assume  that  the 
usual  projective  properties  of  figures  are  true,  since  the 
postulates  of  projective  geometry  are  fully  verified.  Indeed 
the  absence  of  parallels,  or,  what  amounts  to  the  same  thing, 
the  fact  that  two  coplanar  lines  always  intersect,  makes  the 
foundation  of  projectivity  in  elliptic  space  simpler  than  in  Eu- 
clidean space,  which,  as  is  well  known,  must  be  first  com- 
pleted by  the  points  at  infinity. 

However  in  hyperbolic  space  the  matter  is  more  com- 
plicated. Here  it  is  not  sufficient  to  account  for  the  absence 
of  the  point  common  to  two  parallel  lines,  an  exception 
which  destroys  the  validity  of  the  projective  postulate: — two 
coplanar  lines  have  a  coinmon  point.  We  must  also  remove 
the  Other  exception — the  existence  of  coplanar  lines  which 
do  not  cut  each  other,  and  are  not  parallel  {the  non-inter- 
secting lines).  The  method,  which  we  shall  employ,  is  the 
same  as  that  used  in  dealing  with  the  Euclidean  case.  We 
introduce  fictitious  points^  regarded  as  belonging  to  two  co- 
planar lines  which  do  not  meet. 

IS* 


228      ApP-  I^'    ^^^  Indcpend.  of  Proj.  Geo.  from  Euclid's  Post. 

In  the  following  paragraphs,  keeping  for  simplicity  to 
two  dimensions  only,  we  show  how  these  fictitious  points 
can  be  introduced  on  the  hyperbolic  plane,  and  how  they 
enable  us  to  establish  the  postulates  of  projective  geometry 
without  exception.  Naturally  no  distinction  is  now  made  be- 
tween \kit  proper  poitits,  that  is,  the  Ordinary  points,  and  the 
fictitious  points,  thus  introduced. 

§  2.  Improper  Points  and  the  Complete  Projective  Plane. 
We  start  with  the  pencil  of  lines,  that  is,  the  aggregate  of 
the  lines  of  a  plane  passing  through  a  point.  We  note  that 
through  any  point  of  the  plane,  which  is  not  the  vertex  of 
the  pencil^  there  passes  one,  and  only  one,  line  of  the  pencil. 

On  the  hyperbolic  plane,  in  addition  to  the  pencil,  there 
exist  two  other  systems  of  lines  which  enjoy  this  property, 
namely;  — 

(i)  the  set  of  parallels  to  a  line  iti  one  direction', 

(ii)  the  set  of  perpendiculars  to  a  line. 

If  we  extend  the  meaning  of  the  term,  pencil  of  lines, 
we  shall  be  able  to  include  under  it  the  two  systems  of  lines 
above  mentioned.  In  that  case  it  is  clear  that  t7vo  arbi- 
trary lines  of  a  plane  will  determine  a  pencil,  to  7a hie h  they 
belong. 

If  the  two  lines  are  concurrent ,  the  pencil  is  formed  by 
the  set  of  lines  passing  through  their  common  point;  if  they 
are  parallel,  by  the  set  of  parallels  to  both,  in  the  same 
direction;  finally,  if  they  are  nofi-ifitersectifig,  by  all  the  lines 
which  are  orthogonal  to  their  common  perpendicular.  In 
the  first  type  of  pencil  (Ù^e  proper  pencil),  there  exists  a  point 
common  to  all  its  lines,  the  vertex  of  the  pencil;  in  the  two 
other  types  (the  improper  pencils),  this  point  is  lacking.  IVe 
shall  now  introduce,  by  convention,  a  fictitious  entity,  called  an 
improper  point,  and  regard  it  as  pertainitig  to  all  the  lines  of 
the  pencil.    With  this  convention,  every  pencil  has  a  vertex, 


The  Complete  Line  and  Plane.  22Q 

which  will  be  a  proper  point,  or  an  improper  point,  accord- 
ing to  the  different  cases.  The  hyperbolic  plane,  regarded 
as  the  aggregate  of  all  its  points,  proper  and  improper,  will 
be  called  the  complete  projective  plane. 

§  3.  The  Complete  Projective  Line.  The  improper 
points  are  of  two  kinds.  They  may  be  the  vertices  of  pen- 
cils of  parallels,  or  the  vertices  of  pencils  of  non-intersecting 
lines.  The  points  of  the  first  species  are  obtained  in  the 
same  way,  and  have  the  same  use,  as  the  points  at  infinity 
common  to  two  Euclidean  parallels.  For  this  reason  we  shall 
call  them  points  at  infinity  on  the  hyperbolic  plane,  when  it 
is  necessary  to  distinguish  them  from  the  others.  The  points 
of  the  second  species  will  be  called  ideal  points. 

It  will  be  noticed  that,  while  every  line  has  only  one 
point  at  infinity  on  the  Euclidean  plane,  it  has  tivo  points  at 
infinity  on  the  hyperbolic  plane,  there  being  two  distinct 
directions  of  parallelism  for  each  line.  Also  that,  while  the 
line  on  the  Euclidean  plane,  with  its  point  at  infinity,  is 
closed,  the  hyperbolic  line,  regarded  as  the  aggregate  of 
its  proper  points,  and  of  its  two  points  at  infinity,  is  open. 
The  hyperbolic  line  is  closed  by  associating  with  it  all  the 
ideal  points,  which  are  common  to  it  and  to  all  the  lines  on 
the  plane  which  do  not  intersect  it. 

From  this  point  of  view  we  regard  the  line  as  made 
up  of  two  segments.,  whose  common  extremities  are  the  two 
points  at  infinity  of  the  line.  Of  these  segments,  one  contains, 
in  addition  to  its  ends,  all  the  proper  points  of  the  line;  the 
other  all  its  improper  points.  The  line,  regarded  as  the 
aggregate  of  its  points,  proper  and  improper,  will  be  called 
the  complete  projective  line. 

§  4.  Combination  of  Elements.  We  assume  for  the 
concrete  representation  of  a  point  of  the  complete  projective 
plane: — 


2'ZO    App.  IV.     The  Independ.  of  Proj.  Geo.  from  Euclid's.  Post. 

(i)  its  physical  image,  if  it  is  a  proper  point; 

(ii)  a  line  which  passes  through  it,  and  the  relative 
direction  of  the  line,  if  it  is  a  point  at  infinity; 

(iii)  the  common  perpendicular  to  all  the  lines  passing 
through  it,  if  it  is  an  ideal  point. 

We  shall  denote  a  proper  point  by  an  ordinary  capital 
letter;  an  improper  point  by  a  Greek  capital;  and  to  this 
we  shall  add,  for  an  ideal  point,  the  letter  which  will 
stand  for  the  representative  line  of  that  point.  Thus  a  point 
at  infinity  will  be  denoted,  e.  g.,  by  Q,  while  the  ideal  point, 
through  which  all  lines  perpendicular  to  the  line  o  pass,  will 
be  denoted  by  Qo- 

On  this  understanding,  if  we  make  no  distinction  be- 
tween proper  points  and  improper  points,  not  only  can  we 
affirm  the  unconditional  validity  of  the  projective  postulate: 
two  arbitrary  lines  have  a  common  point:  but  we  can  also 
construct  this  point,  understanding  by  this  construction  the 
process  of  obtaining  its  concrete  representation.  In  fact,  if  the 
lines  meet,  in  the  ordinary  sense  of  the  term,  or  are  parallel, 
the  point  can  be  at  once  obtained.  If  they  are  non-inter- 
secting, it  is  sufficient  to  draw  their  common  perpendicular, 
according  to  the  rule  obtained  in  Appendix  III  S  4- 

On  the  other  hand,  we  are  not  able  to  say  that  the 
second  postulate  of  projective  gtovaeXry—tivo  points  determine 
a  line — and  the  corresponding  constructions,  are  valid  un- 
conditionally. In  fact  no  line  passes  through  the  ideal  point 
Qo  and  through  the  point  at  infinity  Q  on  the  line  <?,  since 
there  is  no  line  whicli  is  at  tlie  same  time  parallel  and  per- 
pendicular to  a  line  o. 

Before  indicating  how  we  can  remove  this  and  other 
exceptions  to  the  principle  that  a  line  can  be  determined  by 
a  pair  of  points,  we  shall  enumerate  all  the  cases  in  which 
two  points  fix  a  line,  and  the  corresponding  constructions:  — 

a)  Two  proper  poiiits.    The  line  is  constructed  as  usual. 


Combination  of  Elements. 


231 


b)  A  proper  point  [0]  ajid  a  point  at  infinity  [Q].  The 
line  OQ  is  constructed  by  drawing  the  parallel  through  0  to 
the  line  which  contains  Q,  in  the  direction  corresponding 
to  Q.    (Appendix  III). 

(c)  A  proper  point  [0]  and  an  ideal  point  [PJ.  The  line 
Or^  is  constructed  by  dropping  the  perpendicular  from  0  to 
the  line  c. 

(d)  Two  points  at  infinity  [Q,  Q'].  The  line  QQ'  is  the 
common  parallel  to  the  two  lines  bounding  an  angle,  the 
construction  for  which  is  given  in  Appendix  III  §  5. 

(e)  A71  ideal  point  [fj  and  a  point  at  infinity  [Q],  not 
lying  071  the  representative  lifie  c  of  the  ideal  point.  Tlie  line 
QP^  is  the  line  which  is  parallel  to  the  direction  given  by  Q 
and  perpendicular  to  c.  The  construction  is  given  in  Append- 
ix HI  §  6. 

(f)  Two  ideal  points  [f^ ,  f^'],  whose  representative  lines 
c,  c  do  not  intersect.  The  line  VX  è,  is  constructed  by  drawing 
the  common  perpendicular  to  c  and  /  (Appendix  III  §  4). 

The  pairs  of  points  which  do  not  determine  a  line  are 
as  follows: — 

(i)  an  ideal  point  and  a  point  at  infinity,  lying  on  the 
representative  line  of  the  ideal  point; 

(ii)  two  ideal  points,  whose  representative  lines  are 
parallel,  or  meet  in  a  proper  point. 

§  5.  Itnproper  Lines.  To  remove  the  exceptions  men- 
tioned above  in  (i)  and  (ii),  new  entities  must  be  introduced. 
These  we  shall  call  improper  lines,  to  distinguish  them  firom 
the  ordinary  or  proper  lines. 

These  improper  hnes  are  of  two  types:— 

(i)  If  Q  is  a  point  at  infinity,  every  line  of  the*  pencil  Q 
is  the  representative  entity  of  an  ideal  point.  The  locus  of 
these  ideal  points,  together  with  the  point  Q,  is  an  im- 
proper line  of  the  first  type,  or  line  at  infinity.  It  will  be 
denoted  by  iw. 


2^2      App.  IV.    The  Independ.  of  Proj.  Geo.  from  Euclid's  Post. 

(ii)  If  ^  is  a  proper  point,  every  line  passing  through  A 
is  the  representative  entity  of  an  ideal  point.  The  locus  of 
these  ideal  points  is  an  improper  line  of  the  second  type,  Or 
ù/ea/  line.  It  will  be  denoted  by  a^.  The  proper  point  A 
can  be  taken  as  representative  of  the  ideal  Hne  <1a. 

These  definitions  of  the  terms  line  at  infinity  and  ideal 
li?ie  allow  us  to  state  that  two  points,  which  do  not  belong 
to  a  proper  line,  determine  either  a  line  at  infinity,  or  an 
ideal  line.  Hence,  dropping  the  distinction  between  proper 
and  improper  elements,  the  projective  postulate — two  points 
determine  a  line — is  universally  true. 

We  must  now  show  that,  with  the  addition  of  the  im- 
proper lines,  any  two  lines  have  a  common  point.  The 
various  cases  in  which  the  two  lines  are  proper  have  been 
already  discussed  (§  4).  There  remain  to  be  examined  the 
cases  in  which  at  least  one  of  the  lines  is  improper. 

(i)  Let  r  be  a  proper  hne  and  uj  an  improper  line, 
passing  through  the  point  Q  at  infinity.  The  point  uur  is  the 
ideal  point,  which  has  the  line  passing  through  Q  and  per- 
pendicular to  r  for  representative  line. 

(ii)  Let  r  be  a  proper  line  and  a^  an  ideal  line.  The 
point  ro.A  is  the  ideal  point,  which  has  the  line  passing 
through  A  and  perpendicular  to  r  for  its  representative  line. 

(iii)  Let  UJ  and  uj'  be  two  lines  at  infinity,  to  which 
belong  the  points  Q  and  Q'  respectively.  The  point  ujuj'  is 
the  ideal  point,  whose  representative  line  is  the  line  joining 
the  points  Q  and  Q'. 

(iv)  Let  a,^,  ^B  be  two  ideal  lines.  The  point  o.a'^b  is 
the  ideal  point,  whose  representative  line  is  the  line  joining 
A  and  B, 

(v)  Let  UJ  and  a^  be  a  line  at  infinity  and  an  ideal  hne. 
The  point  uja^  is  the  ideal  point,  whose  representative  line 
is  the  line  joining  ^  to  Q. 

Thus  we  have  demonstrated  that  the  two  fundamental 


Use  of  Improper  Elements.  2^3 

postulates  of  projective  plane  geometry  hold  on  the  hyper- 
bolic plane. 

§  6.  Complete  Projective  Space  a  fid  the  Validity  of  Pro- 
jective Geometry  in  the  Hyperbolic  Space.  We  can  introduce 
improper  points,  lines  and  planes,  into  the  Hyperbolic  Space 
by  the  same  method  which  has  been  followed  in  the  preced- 
ing paragraphs.  We  can  then  extend  the  fundamental  pro- 
positions of  projective  geometry  to  the  complete  projective 
space.  Thereafter,  following  the  lines  laid  down  by  Staudt, 
all  the  important  projective  properties  of  figures  can  be  de- 
monstrated. Thus  the  validity  of  projective  geometry  in  the 
LoBATSCHEWSKY-BoLYAi  Space  is  established. 

§  7.  Indepefidence  of  Projective  Geometry  from  the  Fifth 
Postulate.    Let  us  suppose  that  in  a  connected  argument, 

founded  on  the  group  of  postulates  A,  B ^  H,  the  only 

hypotheses  which  can  be  used  are  /j,  /i ,  /„.    Also  that 

from  the  fundamental  postulates  and  any  one  whatever  of 
the  Is,  a  certain  proposition  M  can  be  derived.  Then  we 
may  say  that  M  is  independent  of  the  I's. 

It  is  precisely  in  this  way  that  the  independence  of  pro- 
jective geometry  from  the  Fifth  Postulate  is  proved,  since 
we  have  shown  that  it  can  be  built  up,  starting  from  the 
group  of  postulates  common  to  the  three  systems  of  geo- 
metry, and  then  adding  to  them  any  one  of  the  hypotheses 
on  parallels. 

The  demonstration  of  the  independence  of  il/ from  any 
one  of  the  I's,  founded  on  the  deduction  (cf  §  59) 

{4^,..    H,Ir)    D     7lf^,=  i,2,...«) 

may  be  called  indirect,  reserving  the  term  direct  demonstration 
for  that  which  shows  that  it  is  possible  to  obtain  AI  without 
introducing  any  of  the  I's  at  all.  Such  a  possibility,  from  the 
theoretical   point   of  view,    is   to   be    expected,   since   the 


2  •34     "''^PP-  ■^^*    ^^^  Independ.  of  Proj.  Geo.  from  Euclid's  Post. 

preceding  relations  show  that  neither  any  single  /,  nor  any 
group  of  them,  is  necessary  to  obtain  M.  If  we  wish  to  give 
a  demonstration  of  the  type 

[A,  B,...Il}^  M, 
in  which  the  /'s  do  not  appear  at  all,  v/e  may  meet  difficult- 
ies not  always  easily  overcome,  difficulties  depending  on  the 
nature  of  the  question,  and  on  the  methods  we  may  adopt 
to  solve  it.  So  far  as  regards  the  independence  of  projective 
geometry  from  the  Fifth  Postulate,  we  possess  two  interesting 
types  of  direct  proofs,  founded  on  two  different  orders  of 
ideas.  One  employs  the  method  of  analysis:  the  other  that 
of  synthesis.  We  shall  now  briefly  describe  the  views  on 
which  they  are  founded. 

§  8.  Beltra7ni's  Direct  Demonstration  of  the  Independ- 
ence of  Projective  Geometry  from  the  Fifth  Postulate.  The 
demonstration  implicitly  contained  in  Beltrami's  ''Saggid  of 
1868  must  be  placed  first  in  chronological  order.  Referring 
to  the  ^Saggio\  let  us  suppose  that  between  the  points  of  a 
surface  F,  {or  of  a  suitably  litnited  region  of  the  surface),  a?id 
the  points  of  an  ordinary  plane  area,  there  can  be  established 
a  one-one  correspondence,  such  that  the  geodesies  of  the  former 
are  represented  by  the  straight  lines  of  the  latter.  Tlien,  to  the 
projective  properties  of  plane  figures,  which  express  the 
collinearity  of  certain  points,  the  concurrence  of  certain 
lines,  etc.,  correspond  similar  properties  of  the  correspond- 
ing figures  on  the  surface,  which  are  deduced  from  the  first, 
by  simply  changing  the  words  platie  and  line  into  surface 
and  geodesic.  If  all  this  is  possible,  we  should  naturally  say 
that  the  projective  properties  of  the  corresponding  plane 
area  are  valid  on  the  surface  F;  or,  more  simply,  that  the 
ordinary  projectivity  of  the  plane  holds  upon  the  surface. 
We  shall  now  put  this  result  in  an  analytical  form. 

Let  u  and  v  be  the  (curvilinear)  coordinates  of  a  point 


Beltrami's  Direct  Demonstration. 


235 


on  F^  and  x  and  y  those  of  the  representative  point  on  the 
plane.    The  correspondence  between  the  points  {u^  v)  and 
{x,  y)  will  be  expressed  analytically  by  putting 
u  ^  f  {x,  y)\ 


z'  =  cp  (^,  y)\ 
where  y^  and  qp  are  suitable  functions. 

To  the  equation 

ip  {u,  v)  ^  0 
of  a  geodesic  on  F,  let  us  now  apply  the  transformation  (i). 

We  must  obtain  a  linear  equation  in  x,  y,  since,  by  our 
hypothesis,  the  geodesies  of  F  are  represented  by  straight 
lines  on  the  plane. 

But  the  equations  (i)  can  also  be  interpreted  as  formulae 
giving  a  transformatiofi  of  coordinates  on  F.  We  can  there- 
fore conclude  that: — If,  by  a  suitable  choice  of  a  system  of 
curvilinear  coordinates  on  the  surface  F,  the  geodesies  of  that 
surface  can  be  represented  by  linear  equations,  the  ordinary 
projective  geometry  is  valid  on  the  surface. 

Now  Beltrami  has  shown  in  his  ^Saggio'  that  on  surfaces 
of  constant  curvature  it  is  always  possible  to  choose  a  system 
of  coordinates   (u,  v),  for  which  the  general  integral  of  the 
differential  equation  of  the  geodesies  takes  the  form 
ax  +  by  +  c  =^  0. 

Hence,  from  what  has  been  said  above,  it  follows  that: — 

Plane  projective  geometry  is  valid  on  the  surfaces  of  con- 
stant curvature  with  respect  to  their  geodesies. 

But,  according  to  the  value  of  the  curvature,  the  geo- 
metry of  these  surfaces  coincides  with  that  of  the  Euclidean 
plane,  or  of  the  Non-Euclidean  planes. 

It  follows  that: — 

The  method  of  Beltrami,  applied  to  a  plane  on  which  are 
valid  the  metrical  concepts  co?n?non  to  the  three  geometries,  leads 


2 ■36     ApP'  IV.    The  Independ.  of  Proj.  Geo.  from  Euclid's  Post. 

to    the  foundation   of  pla?ie  projective  geometry   without  the 
assumption  of  any  hypothesis  on  parallels. 

This  result  and  the  argument  we  have  employed  in  ob- 
taining it  are  easily  extended  to  space.  Beltrami's  memoir 
referring  to  this  is  the  Teoria  fondamentale  degli  spazii  di 
curvatura  costa?ite,  quoted  in  the  note  to  §  75. 

§  9.  Klein's  Direct  Demonstration  of  the  Independence 
of  Projective  Geometry  from  the  Fifth  Fostulate.  The  method 
indicated  above  is  not  the  only  one  which  will  serve  our 
purpose.  In  fact,  we  might  be  asked  if  we  could  not  construct 
projective  geometry  independently  of  any  metrical  consider- 
ation; that  is,  starting  from  the  notions  of  point,  line,  plane, 
and  from  the  axioms  of  connection  and  order,  and  the  prin- 
ciple of  continuity.'  In  187 1  Klein  was  convinced  of  the 
possibility  of  such  a  foundation,  from  the  consideration  of 
the  method  followed  by  Staudt  in  the  construction  of  his 
geometrical  system.  There  remained  one  difficulty,  relative 
to  the  improper  points.  Staudt,  following  Poncelet,  makes 
them  to  depend  on  the  ordinary  theory  of  parallels.  To 
escape  the  various  exceptions  to  the  statement  that  two 
coplanar  lines  have  a  common  point,  due  to  the  omission  of 
the  Euclidean  hypothesis,  YiLEm  proposed  to  construct  projective 
geometry  in  a  limited  {and  convex)  region  of  space,  such,  e.  g., 
as  that  of  the  points  inside  a  tetrahedron.  With  reference  to 
such  a  region,  for  the  end  he  has  in  view,  every  point  on 
the  faces  of,  or  external  to,  the  tetrahedron  must  be  con- 
sidered as  non-existent.  Also  we  must  give  the  name  of  line 
and  plane  only  to  the  portions  0/ the  line  and  plane  belonging 
to  the  region  considered.  Then  the  graphical  postulates  of 
connection,  order,  etc.,  which  are  supposed  true  in  the  whole 


I  For  this  nomenclature  for  the  Axioms,  cf.  Tuwnsend's 
translation  of  Hilbert's  Fowtdatioiis  of  Geometiy,  p.  I  (Open  Court 
Publishing  Co.   1902). 


Klein's  Direct  Demonstration.  23/ 

of  space,  are  verified  in  the  interior  of  the  tetrahedron.  Thus 
to  construct  projective  geometry  in  this  region,  it  is  neces- 
sary, with  suitable  conventions,  that  the  propositions  on  the 
concurrence  of  hues,  etc.  should  hold  without  exception. 
These  are  not  always  true,  when  the  word  point  means 
simply  point  inside  the  tetrahedron. 

Klein  showed  briefly,  while  various  later  writers  dis- 
cussed the  question  more  fully,  how  the  space  inside  the 
tetrahedron  can  be  completed  by  fictitious  entities,  called 
ideal  points,  lines  and  planes,  so  that  when  no  distinction 
is  made  between  the  proper  entities  (inside  the  tetrahedron) 
and  the  ideal  entities,  the  graphical  properties  of  space,  on 
which  all  projective  geometry  is  constructed,  are  completely 
verified. 

From  this  there  readily  follows  the  independence  of 
projective  geometry  from  Euclid's  Fifth  Postulate. 


Appendix  V. 

The  Impossibility  of  Proving  Euclid's 
Parallel  Postulate.' 

An  Elementary  Demonstration  of  this  Impossibility  founded 
upon  the  Properties  of  the  System  of  Circles  orthogonal  to  a 
Fixed  Circle. 

§  I.  In  the  concluding  article  (§  94)  various  arguments 
are  mentioned,  any  one  of  which  establishes  the  independence 
of  Euclid's  Parallel  Postulate  from  the  other  assumptions  on 
which  Euclidean  Geometry  is  based.  One  of  these  has  been 
discussed  in  greater  detail  in  Appendix  IV.  In  the  articles 
which  follow  there  will  be  found  another  and  a  more  ele- 
mentary proof  that  the  Bolyai-Lobatschewsky  system  of 
Non-Euclidean  Geometry  cannot  lead  to  any  contradictory 
results,  and  that  it  is  therefore  impossible  to  prove  Euclid's 
Postulate  or  any  of  its  equivalents.  This  proof  depends,  for 
solid  geometry,  upon  the  properties  of  the  system  of  spheres  all 
orthogonal  to  a  fixed  sphere,  while  for  plane  geometry  the 
system  of  circles  all  orthogonal  to  a  fixed  circle  is  taken. 
In  the  course  of  the  discussion  many  of  the  results  of  Hyper- 
bolic Geometry  are  deduced  from  the  properties  of  this 
system  of  circles. 


I  This  Appendix,  added  to  the  English  translation,  is  based 
upon  Wellstein's  work,  referred  to  on  p.  I  So,  and  the  following 
paper  by  Carslaw;  ^The  Bolyai-Lobatschewsky  Non-EiicUdeaii  Geo- 
metry: an  Elementary  Interpretation  of  this  Geometry  and  some  Results 
which  follow  from  this  Interpretation,  Proc.  Edin.  Math.  See.  Vol. 
XXVIII,  p.  95  (1 910). 

Cf.  also  :  J.  WellstEIN,  Zusammeiihang  zwischen  zwei  euklid- 
tscheft  Bilderit  der  nichtei/klidischcn  Geometric.  Archiv  der  Math.  u. 
Physik  (3).  XVII,  p.  19s  (1910). 


Ideal  Lines. 


239 


The  System  of  Circles  passing  through  a  fixed  Point. 

§  2.  We  shall  examine  first  of  all  the  representation  of 
ordinary  Euclidean  Geometry  by  the  geometry  of  the  system 
of  spheres  all  passing  through  a  fixed  point.  In  plane  geo- 
metry this  reduces  to  the  system  of  circles  through  a  fixed 
point,  and  we  shall  begin  with  that  case. 

Since  the  system  of  circles  through  a  point  O  is  the 
inverse  of  the  system  of  straight  lines  lying  in  the  plane,  to 
every  circle  there  corresponds  a  straight  line,  and  the  circles 
intersect  at  the  same  angle  as  the  corresponding  hnes.  The 
properties  of  the  set  of  circles  could  be  established  from  the 
knowledge  of  the  geometry  of  the  straight  lines,  and  every 
proposition  concerning  points  and  straight  lines  in  the  one 
geometry  could  at  once  be  interpreted  as  a  proposition  con- 
cerning points  and  circles  in  the  other. 

There  is  another  way  in  which  the  geometry  of  these 
circles  can  be  established  independently.  We  shall  first  de- 
scribe this  method,  and  weshall  then  see  that  from  this  inter- 
pretation of  the  Euclidean  Geometry  we  can  easily  pass  to  a 
corresponding  representation  of  the  Non-Euclidean  Geometry. 

§  3.   Ideal  Lines. 

It  will  be  convenient  to  speak  ot  the  plane  of  the 
straight  lines  and  the  plane  of  the  circles,  as  two  separate 
planes.  We  have  seen  that  to  every  straight  line  in  the  plane 
of  the  straight  lines,  there  corresponds  a  circle  in  the  plane 
of  the  circles.  We  shall  call  these  circles  Ideal  Lines.  The 
Ideal  Points  will  be  the  same  as  ordinary  points,  except  that 
the  point  O  will  be  excluded  from  the  domain  of  the  Ideal 
Points. 

On  this  understanding  we  can  say  that  Any  two  different 
Ideal  Points,  A,  B,  determine  the  Ideal  Line  A£;  just  as,  in 
Euclidean  Geometry,  any  two  different  points  A,  B  deter- 
mine the  straight  line  AB. 


240      Appendix  V.    Impossibility  of  proving  Euclid's  Postulate. 

As  the  angle  between  the  circles  in  the  one  plane  is 
equal  to  the  angle  between  the  corresponding  straight  lines 
in  the  other,  we  define  the  angle  between  tivo  Ideal  Lines  as 
the  angle  between  the  corresponding  straight  lines.  Thus  we 
can  speak  of  Ideal  Lines  being  perpendicular  to  each  other, 
or  cutting  at  any  angle. 

§  4.    Ideal  Parallel  Lines. 

Let  BC  (cf.  Fig.  83)  be  any  straight  line  and  A  a  point 
not  lying  upon  it. 


Let  AM  be  the  perpendicular  to  BC,  and  AM^ ,  AM^, 
AAf,,  .  .  .  different  positions  of  the  line  AM,  as  it  revolves 
from  the  perpendicular  position  through  two  right  angles. 

The  lines  begin  by  cutting  BC  on  the  one  side  of  Af, 
and  there  is  one  line  separating  the  lines  which  intersect 
BC  on  the  one  side,  from  those  which  intersect  it  on  the 
other.    This  line  is  the  parallel  through  A  to  BC. 

In  the  corresponding  figure  for  the  Ideal  Lines  (cf. 
Fig.  84),  we  have  the  Ideal  Line  through  A  perpendicular  to 
the  Ideal  Line  BC;  and  the  circle  which  passes  through  A, 
and  touches  the  circle  OBC  at  O,  separates  the  circles 
through  A,  which  cut  BC  on  the  one  side  of  31,  from  those 
which  cut  it  on  the  other. 


Ideal  Parallels. 


241 


We  are  thus  led  to  define  Parallel  Ideal  Lines  as  follows: 

The  Ideal  Line  through  a?iy  point  parallel  to    a  given 

Ideal  Line  is  the  circle  of  the  system  which  touches  at  O  the 

circle  coinciding  with  the  given  line  and  also  passes  through  the 

given  point. 


Thus  any  two  circles  of  the  system  which  touch  each 
other  at  O  will  be  Ideal  Parallel  Lines.  Two  Ideal  Lines, 
which  are  each  parallel  to  a  third  Ideal  Line,  are  parallel  to 
each  other,  etc. 

§  5.    Ideal  Leiigths. 

Since  Euclid's  Parallel  Postulate  is  equivalent  to  the 
assumption  that  one,  and  only  one,  straight  line  can  be 
drawn  through  a  point  parallel  to  another  straight  line,  and 
since  this  postulate  is  obviously  satisfied  by  the  Ideal  Line, 

16 


242      Appendix  V.     Impossibility  of  proving  Euclid's  Postulate. 

in  the  geometry  of  these  Hnes,  Euclid's  Theory  of  Parallels 
will  be  true. 

But  such  a  geometry  will  require  a  measurement  of 
length.  We  must  now  define  what  is  meant  by  the  Ideal 
Lmgth  of  an  Ideal  Segment.  In  other  words  we  must  define 
the  Ideal  Distance  between  two  points.  It  is  clear  that  if  the 
two  geometries  are  to  be  identical  two  Ideal  Segments  must 
be  regarded  as  of  equal  length,  when  the  corresponding 
rectilinear  segments  are  equal.  We  thus  define  the  Ideal 
Length  of  an  Ideal  Segment  as  the  length  of  the  rectilinear 
segmmt  to  which  it  corresponds. 

It  will  be  seen  that  the  Ideal  Distance  between  two 
points  y^,  B  is  such  that,  if  C  is  any  other  point  on  the 
segment, 

'distance'  AB  =  'distance'  AC  ^  'distance'  CB. 

The  other  requisite  for  'distance'  is  that  it  is  unaltered 
by  displacement,  and  when  we  come  to  define  Ideal  Dis- 
placement we  shall  have  to  make  sure  that  this  condition  is 
also  satisfied. 

It  is  clear  that  on  this  understanding  the  Ideal  Length 
of  an  Ideal  Line  is  infinite.  If  we  take  'equal*  steps  along 
the  Ideal  Line  BC  from  the  foot  of  the  perpendicular  (cf. 

Fig.  84)  the  actual  lengths  of  the  arcs  MMi ,  M^M^,  etc , 

the  Ideal  Lengths  of  which  are  equal,  become  gradually 
smaller  and  smaller,  as  we  proceed  along  the  line  towards  O. 
It  will  take  an  infinite  number  of  such  steps  to  reach  O,  just 
as  it  will  take  an  infinite  number  of  steps  along  BC  from  AI 
(cf  Fig.  83)  to  reach  the  point  at  which  BC  is  met  by  the 
parallel  through  A.  We  have  already  seen  that  the  domain 
of  Ideal  Points  contains  aU  the  points  of  the  plane  except 
O.  This  was  required  so  that  the  Ideal  Line  might  always 
be  determined  by  two  different  points.  It  is  also  needed  for 
the  idea  of  'between-ness'.  On  the  straight  line  AB  we.  can 
say  that  C  lies  between  line  A  and  B  if,  as  we  proceed  along 


Ideal  Lengths. 


243 


AB  from  A  to  B^  we  pass  through  C.  On  the  Ideal  Line  AB 
(cf.  Fig.  85)  the  points  G  and  C2  would  both  lie  between 
A  and  B,  unless  the  point  O  were  excluded.  In  other  words 
this  convention  must  be  made  so  that  the  Axioms  of  Order  ^ 
may  appear  in  the  geometry  of  the  Ideal  Points  and  Lines. 


Fig   85. 

On  this  understanding,  and  still  speaking  of  plane  geo- 
metry, we  can  say  that  two  Ideal  Lines  are  parallel  when  they 
do  not  meet,  however  far  they  are  produced. 

To  obtain  an  expression  for  the  Ideal  Length  of  an 
Ideal  Segment  we  may  take  the  radius  of  inversion — k — to 
be  unity. 

Consider  the  segment  AB  and  the  rectihnear  segment 
aP  to  which  it  corresponds.  Then  we  have  (Fig.  86) 
^P   _  Op  _  op.  OB  _  _^2 
AB  ~  OA  ^  OA.  OB  ~  ÒA~.0B' 


I  See  Note  on  p.  236. 


16* 


244     Appendix  V.     Impossibility  of  proving  Euclid's  Postulate. 

Hence  we  define  the  Ideal  Length  of  the  segf>tent  AB  as 
AB 


OA.  OB 
We  shall  now  show  that  the  Ideal  length  of  an  Ideal 
Segment  is  unaltered  by  inversivi  with  regard  to  any  circle  of 
the  system. 


Fig.  So. 

Let  OD  be  any  circle  of  the  system  and  let  C  be  its 
centre  (Fig.  87). 

Then  inversion  changes  an  Ideal  Line  into  an  Ideal 
Line. 

Let  the  Ideal  Segment  AB  invert  into  the  Ideal  Segment 
A'B'.    These  two  Ideal  Lines  intersect  at  the  point  D,  where 
the  circle  of  inversion^meets  AB. 
Then 

the  Ideal  Length  of  AD   AD      1     A'D 

the  Ideal  Length  of ^'^  ~  OA.  ODj  OA' .  OD 
__  AD       OA 
~'  'ad  '    OA' 

But  from  the  triangles  CAD,  CAD  and  OAC,  OA'C, 
we  find 


Ideal  Displacements. 


245 


AD 


CA 
CD 


CA 
CO 


AO 
A^O' 


Thus  the  Ideal  Length  oi  AD  =  the  Ideal  Length  oiA'D. 
Similarly  we  find  BD  and  B'D  have  the  same  Ideal  Length, 
and  therefore  AB  and  A'B'  have  the  same  Ideal  Length. 


Fig.  87. 

§  6.    Ideal  Displacements. 

The  length  of  a  segment  must  be  unaltered  by  dis- 
placement. This  leads  us  to  consider  the  definition  of  Ideal 
Displacement.  Any  displacement  may  be  produced  by  re- 
peated applications  of  reflection;  that  is,  by  taking  the  image 
of  the  figure  in  a  line  (or  in  a  plane,  in  the  case  of  solid 
geometry).  For  example,  to  translate  the  segment  AB  (cf. 
Fig.  88)  into  another  position  on  the  same  straight  line,  we 


246     Appendix  V.     Impossibility  of  proving  Euclid's  Postulate. 

may  reflect  the  figure,  first  about  a  line  perpendicular  to  and 
bisecting  BB' ,  and  then  another  reflection  about  the  middle 
point  of  AB  would  bring  the  ends  into  their  former  positions 
relative  to  each  other.    Also  to  move  the  segment  AB  into 


A 


B 


B' 


— I 
A' 


Fig. 


the  position  AB'  (cf.  Fig.  89)  we  can  first  take  the  image  ot 
AB  in  the  line  bisecting  the  angle  between  AB  and  AB\ 
and  then  translate  the  segment  along  AB'  to  its  final 
position. 

We  proceed  to  show 
that  inversion  about  any 
circle  of  the  system  is 
equivalent  to  reflection  of 
the  Ideal  Points  and  Lines 
in  the  Ideal  Line  which 
coincides  with  the  circle 
of  Ì7iversion. 

Let  C  (Fig.  90)  be 
the  centre  of  any  circle 
of  the  system,  and  let  A 
be  the  inverse  of  any 
point  A  with  regard  to 
this  circle.  Then  the 
circleO AA'  is  orthogonal 
to  the  circle  of  inversion. 
In  other  words,  such  inversion  changes  any  point  A  into  a 
point  A  on  the  Ideal  Line  perpendicular  to  the  circle  of  in- 
version. Also  the  Ideal  Line  AA  is  'bisected'  by  that  circle 
at  M,  since  the  Ideal  Segment  AM  inverts  into  the  segment 
AM,  and  Ideal  Lengths  are  unaltered  by  such  inversion. 

Again  let  AB  be  any  Ideal  Segment,  and  by  inversion 


Fig.  89. 


Ideal  Reflection. 


247 


with  regard  to  any  circle  of  the  system  let  it  take  up  the 
position  AS  (Fig.  8  7).  We  have  seen  that  the  Ideal  Length 
of  the  segment  is  unaltered:  and  it  is  clear  that  the  two 
segments,  when  produced,  meet  on  the  circle  of  inversion, 
and  make  equal  angles  with  it.    Also  the  Ideal  Lines  A  A 


Fig.  90. 

and  BB'  are  perpendicular  to,  and  'bisected'  by,  the  Ideal 
Line  with  which  the  circle  of  inversion  coincides. 

Such  an  inversion  is,  therefore,  the  same  as  reflection, 
and  translation  will  occur  as  a  special  case  of  the  above, 
when  the  circle  of  inversion  is  orthogonal  to  the  given 
Ideal  Line. 

We  thus  define  Ideal  Reflection  m  an  Ideal  Line  as  in- 
version with  this  line  as  the  circle  of  inversion. 

It  is  unnecessary  to  say  more  about  Ideal  Displace- 
fne?its  than  that  they  will  be  the  result  of  Ideal  Reflection. 

With  these  definitions  it  is  now  possible  to  'translate' 
every   proposition  in  the   ordinary  plane  geometry  into  a 


248     Appendix  V.     Impossibility  of  proving  Euclid's  Postulate. 

corresponding  proposition  in  this  Ideal  Geometry.  We  have 
only  to  use  the  words  Ideal  Points,  Lines,  Parallels,  etc., 
instead  of  the  ordinary  points,  lines,  parallels,  etc.  The 
argument  employed  in  proving  a  theorem,  or  the  con- 
struction used  in  solving  a  problem,  will  be  applicable, 
word  for  word,  in  the  one  geometry  as  well  as  in  the  other, 
for  the  elements  involved  satisfy  the  same  laws.  This  is  the 
'dictionary'  method  so  frequently  adopted  in  the  previous 
pages  of  this  book. 

§  7.  Extension  to  Solid  ^  Geometry.  The  System  of 
Spheres  passing  through  a  fixed  point. 

These  methods  may  be  extended  to  solid  geometry.  In 
this  case  the  inversion  of  the  system  of  points,  lines,  and 
planes  gives  rise  to  the  system  of  points,  circles  intersecting 
in  the  centre  of  inversion,  and  spheres  also  intersecting  in 
that  point.  The  geometry  of  this  system  of  spheres  could  be 
derived  from  that  of  the  system  of  points^  lines  and  planes, 
by  interpreting  each  proposition  in  terms  of  the  inverse 
figures.  For  our  purpose  it  is  better  to  regard  it  as  derived 
from  the  former  by  the  invention  of  the  terms:  Ideal 
Point,  Ideal  Line,  Ideal  Plane,  Ideal  Length  and  Ideal  Dis- 
placement. 

The  Ideal  Point  is  the  same  as  the  ordinary  point,  but 
the  point  O  is  excluded  from  the  domain  of  Ideal  Points. 

The  Ideal  line  through  two  Ideal  Points  is  the  circle  of 
the  system  which  passes  through  these  two  points. 

The  Ideal  Flafie  through  three  Ideal  Points,  not  on  an 
Ideal  Line,  is  the  sphere  of  the  system  which  passes  through 
these  three  points. 

Thus  the  plane  geometry,  discussed  in  the  preceding 
articles,  is  a  special  case  of  this  plane  geometry. 

Ideal  Parallel  Lines  are  defined  as  before.  The  line 
through  A  parallel  to  ^C  is  the  circle  of  the  system,  lying 


Extension  to  Solid  Geometry.  249 

on  the  sphere  through  O,  Ay  B,  and  C,  which  touches  the 
circle  given  by  the  Ideal  Line  .BC  at  O  and  passes  through  A. 

It  is  clear  that  an  Ideal  Line  is  determined  by  two 
points,  as  a  straight  line  is  determined  by  two  points.  An 
Ideal  Plane  is  determined  by  three  points,  not  on  an  Ideal 
Line,  as  an  ordinary  plane  is  determined  by  three  points, 
not  on  a  straight  line.  If  two  points  of  an  Ideal  Line  lie  on 
an  Ideal  Plane,  all  the  points  of  the  line  do  so  :  just  as  if  two 
points  of  a  straight  line  lie  on  a  plane,  all  its  points  do  so. 
The  intersection  of  two  Ideal  Planes  is  an  Ideal  Line;  just  as 
the  intersection  of  two  ordinary  planes  is  a  straight  line. 

The  measurement  of  angles  in  the  two  spaces  is  the  same. 

For  the  measurement  of  length  we  adopt  the  same  de- 
finition of  Ideal  Length  as  in  the  case  of  two  dimensions. 
The  Ideal  Length  of  an  Ideal  Segment  is  the  length  of  the 
rectilinear  segment  to  which  it  corresponds.  To  these  defi- 
nitions it  only  remains  to  add  that  of  Ideal  Displacement. 
As  in  the  two  dimensional  case,  this  is  reached  by  means  of 
Ideal  Reflection  :  and  it  can  easily  be  shown  that  if  the  system 
of  Ideal  Poitits,  Lines  and  Planes  is  inverted  with  regard  to 
one  of  its  spheres,  the  result  is  equivalent  to  a  reflection  of  the 
system  in  this  Ideal  Platie. 

This  Ideal  Geometry  is  identical  with  the  ordinary 
Euclidean  Geometry.  Its  elements  satisfy  the  same  laws: 
every  proposition  vaUd  in  the  one  is  also  valid  in  the  other: 
and  from  the  results  of  Euclidean  Geometry  those  of  the 
Ideal  Geometry  can  be  inferred. 

In  the  articles  that  follow  we  shall  establish  an  Ideal 
Geometry  whose  elements  satisfy  the  axioms  upon  which  the 
Non-Euclidean  Geometry  of  Bolyai-Lobatschewsky  is  based. 
The  points,  lines  and  planes  of  this  geometry  will  be  figures 
of  the  Euclidean  Geometry,  and  from  the  known  properties 
of  these  figures,  we  could  state  what  the  corresponding  the- 
orems of  this  Non-Euclidean  Geometry  would  be.    Also  from 


2  co     Appendix  V.     Impossibility  of  proving  Euclid's  Postulate. 

some  of  its  constructions,  the  Non-Euclidean  constructions 
could  be  obtained.  This  process  would  be  the  converse  of 
that  referred  to  in  dealing  with  the  Ideal  Geometry  of  the 
preceding  articles;  since,  in  that  case,  we  obtained  the  the- 
orems of  the  Ideal  Geometry  from  the  corresponding  Eu- 
clidean theorems. 

The  Geometry  of  the  System  of  Circles  Orthogonal 
to  a  Fixed  Circle. 

§  8.    Ideal  Poiiiis,  Ideal  lines  and  Ideal  Parallels. 

In  the  Ideal  Geometry  discussed  in  the  previous  articles, 
the  Ideal  Point  was  the  same  as  the  ordinary  point,  and  the 
Ideal  Lines  and  Planes  had  so  far  the  characteristics  of 
straight  lines  and  planes  that  they  were  lines  and  surfaces 
respectively.  Geometries  can  be  constructed  in  which  the 
Ideal  Points,  Lines  and  Planes  are  quite  rem.oved  from 
ordinary  points,  lines,  and  planes:  so  that  the  Ideal  Points 
no  longer  have  the  characteristic  of  having  no  parts:  and 
the  Ideal  Lines  no  longer  boast  only  length,  etc.  What  is 
required  in  each  geometry  is  that  the  entities  concerned 
satisfy  the  axioms  which  form  the  foundations  of  geometry. 
If  they  satisfy  the  axioms  of  Euclidean  Geometry,  the  argu- 
ments, which  lead  to  the  theorems  of  that  geometry,  will 
give  corresponding  theorems  in  the  Ideal  Geometry:  and  if 
they  satisfy  the  axioms  of  any  of  the  Non-Euclidean  Geom- 
etries, the  arguments^  which  lead  to  theorems  in  that  Non- 
Euclidean  Geometry,  will  lead  equally  to  theorems  in  the 
corresponding  Ideal  Geometry. 

We  proceed  to  discuss  the  geometry  of  the  system  of 
circles  orthogonal  to  a  fixed  circle. 

Let  the  fundamental  circle  be  of  radius  k  and  centre  O. 

Let  A,  A"  be  any  two  inverse  points,  A  being  inside 
the  circle.  Every  such  pair  of  points  {A,  A'),  is  an  Ideal 
Point  {A)  of  the  Ideal  Geometry  with  which  we  shall  71010  deal. 


Circles  orthogonal  to  a  fixed  Circle. 


251 


If  two  such  pairs  of  points  are  given — that  is,  two  Ideal 
Points  (A,  B),  (Fig.  92) — these  determine  a  circle  which  is 
orthogonal  to  the  fundamental  circle.  Every  such  circle  is 
a?i  Ideal  Line  of  this  Ideal  Geometry. 


Fig.  91. 

Hence  any  two  different  Ideal  Points  determine  an  Ideal 
Line.  In  the  case  of  the  system  of  circles  passing  through  a 
fixed  point  O,  this  point  O  was  excluded  from  the  domain 
of  the  Ideal  Points.  In  this  system  of  circles  all  orthogonal 
to  the  fundamental  circle,  the  coincident  pairs  of  points  lying 
on  the  circumference  of  that  circle  are  excluded  from  the 
domain  of  the  Ideal  Points. 

We  define  the  angle  between  two  Ideal  Lines  as  the  angle 
between  the  circles  which  coincide  with  these  lines. 

We  have  now  to  consider  in  what  way  it  will  be  proper 
to  define  Parallel  Ideal  Lines. 

Let  AàI  be  the  Ideal  Line  through  A,  perpendicular  to 
the  Ideal  Line  BC;  in  other  words,  the  circle  of  the  system 
passing  through  A',  A'\  and  orthogonal  to  the  circle  through 
£',  B",  C  and  C"  (cf  Fig.  92). 


2^2     Appendix  V.     Impossibilty  of  proving  Euclid's  Postulate. 

Imagine  AM  to  rotate  about  A  so  that  those  Ideal 
Lines  through  A  cut  the  Ideal  Line  BC  at  a  gradually 
decreasing  angle.   The  circles  through  A  which  touch  the  given 


Fig.  92. 

circle  £C  at  the  points  [/,  V,  where  it  meets  the  fundamental 
circle,  are  Ideal  Lines  of  the  system.  They  separate  the 
lines  of  the  pencil  of  Ideal  Lines  through  A,  which  cut  the 
Ideal  Line  -BC,  from  those  which  do  not  cut  that  line.  All 
the  lines  in  the  angle  q),  shaded  in  the  figure,  do  not  cut 
the  line  £C;  all  those  in  the  angle  ^),  not  shaded,  do  cut 
this  line.  This  property  is  exactly  what  is  assumed  in  the 
Parallel  Postulate  upon  which  the  Non-Euclidean  Geometry 
of  BoLYAi-LoBATSCHEWSKY  is  based.  We  therefore  are  led  to 
define  Parallel  Ideal  Lines  in  this  Plane  Ideal  Geometry  as 
follows: 

TAe  Ideal  Lines  through  an  Ideal  Point  parallel  to  a 
given  Ideal  Line  are  the  two  circles  of  the  syston  passing 


Some  Theorems  in  this  Geometry.  253 

through  the  given  pointy  which  touch  the  circle  with  7vhich  the 
given  line  coincides  at  the  poi?tts  where  it  meets  the  fundam- 
ental circle. 

Thus  we  have  in  this  Ideal  Geometry  two  parallels 
through  a  point  to  a  given  line:  a  right-handed  parallel,  and 
a  left-handed  parallel:  and  these  separate  the  lines  of  the 
pencil  which  intersect  the  given  line  from  those  which  do 
not  intersect  it. 

Some  Theorems  of  this  Non-Euclidean  Geometry. 

§  9.  At  this  stage  we  can  say  that  any  of  the  theorems 
of  the  BoLYAi-LoBATSCHEwsKY  Non-EucHdean  Geometry,  in- 
volving angle  properties  only,  will  hold  in  this  Ideal  Geo- 
metry and  vice  versa.  Those  involving  lengths  we  cannot  yet 
discuss,  as  we  have  not  yet  defined  Ideal  Lengths.  For 
example,  it  is  obvious  that  there  are  triangles  in  which  all 
the  angles  are  zero  (cf.  Fig.  93).  The  sides  of  such  triangles 
are  parallel  in  pairs.  Thus  the  sum  of  the  angles  of  an  Ideal 
Triangle  is  certainly  not  always  equal  to  two  right  angles. 
We  can  prove  that  this  sum  is  always  less  than  two  right 
angles  by  a  simple  application  of  inversion,  as  follows: 

Let  Ci,  C2,  C3  be  three  circles  of  the  system,  forming 
an  Ideal  Triangle.  Invert  these  circles  from  the  point  of 
intersection  /  of  C^  and  C2 ,  which  hes  inside  the  fundament- 
al circle.  Then  d  and  C2  become  two  straight  lines  d' 
and  C2'  through  /.  Also  the  fundamental  circle  C  inverts 
into  a  circle  C  cutting  Ci  and  C2  at  right-angles,  so  that 
its  centre  is  /.  Again,  the  circle  C.  inverts  into  a  circle  C3', 
cutting  C  at  right-angles.  Hence  its  centre  lies  outside  C. 
We  thus  obtain  a  'triangle',  in  which  the  sum  of  the  angles 
is  less  than  two  right-angles,  and  since  these  angles  are  equal 
to  the  angles  of  the  Ideal  Triangle,  this  result  holds  also  for 
the  Ideal  Triangle. 


2^4     Appendix  V.     Impossibility  of  proviug  Euclid's  Postulate. 

Finally,  it  can  be  shown  that  there  is  always  one,  and 
only  one,  circle  of  the  system  cutting  two  non-intersecting 
circles  of  the  system   at  right-angles.    In  other  words,  two 


Fig.  93- 

non-intersecting  Ideal  Lines  have  a  common  perpendicular. 
All  these  results  must  be  true  in  the  Hyperbolic  Geometry. 

§  IO.    Ideal  Lengths  and  Ideal  Displacements. 

Before  we  can  proceed  to  the  discussion  of  the  metrical 
properties  of  this  geometry,  we  must  define  the  Ideal  Length 
of  an  Ideal  Segment.  It  is  clear  that  this  must  be  such  that 
it  will  be  unaltered,  if  we  take  the  points  A\  B",  as  defining 
the  segment  AB,  instead  of  the  points  A\  B'.  It  must  make 
the  complete  line  infinite  in  length.  It  must  satisfy  the  distri- 
butive law  'distance'  AB  =  'distance'  AC  -\-  'distance'  CB, 


Ideal  Lengths. 


255 


if  C  is  any  other  point  on  the  segment  AB^   and  it  must 
also  remain  \in.di\i&xQàhy  Ideal  Displacement. 

We  defaie  the  Ideal  length  of  a?iy  segment  AB  as 

'V_  I  B'^\ 
77/  WI/J 

where  U,  V  are  the  points  where  the  Ideal  Line  AB  meets  the 

fundamental  circle  (cf.  Fig.  91). 


lot 


\a'i 


Fig.  94. 

This  expression  obviously  involves  the  Anharmonic 
Ratio  of  the  points  UABV.  It  will  be  seen  that  this  de- 
finition satisfies  the  first  three  of  the  conditions  named  above. 
It  remains  for  us  to  examine  what  must  represent  dis- 
placement in  this  Ideal  Geometry. 

Let  us  consider  what  is  the  effect  of  inversion  with 
regard  to  a  circle  of  the  system  upon  the  system  of  Ideal 
Points  and  Lines. 

Let  A  A"  be  any  Ideal  Point  A  (cf.  Fig.  94).   Let  the 


2CS     Appendix  V.     Impossibility  of  proving  Euclid's  Postulate. 

circle  of  inversion  meet  the  fundamental  circle  in  C,  and  let 
D  be  its  centre.  Let  A',  A"  invert  into  B',  B" .  Since  the 
circle  A  A'  C  touches  the  circle  of  inversion  at  C,  its  inverse 
also  touches  that  circle  at  C.  But  a  circle  passes  through 
A ^  A",  B'  and  B'\  and  the  radical  axes  of  the  three  circles 

AA'C,  B'B"C,  AA'B'B" 
are  concurrent. 

Hence  B' B"  passes  through  6>,  and  OB' .  OB"  =  0C\ 

Therefore  inversion  with  regard  to  any  circle  of  the 
system  changes  an  Ideal  Point  into  an  Ideal  Point. 

But  it  is  clear  that  the  circle  AA'B'B"  is  orthogonal  to 
the  fundamental  circle,  and  also  to  the  circle  of  inversion. 

Thus  the  Ideal  Line  joining  the  Ideal  Point  A  and  the 
Ideal  Point  B,  into  which  it  is  changed  by  this  inversion,  is 
perpendicular  to  the  Ideal  Line  coincidiiig  with  the  circle  of 
itwersion. 

We  shall  now  prove  that  it  is  'bisected'  by  that  Ideal 
Line. 

Let  the  circle  through  AB  meet  the  circle  of  inversion 
at  M,  and  the  fundamental  circle  in  U  and  V.  It  is  clear 
that  U  and  V  are  inverse  points  with  regard  to  the  circle  of 
inversion  [cf.  Fig.  95]. 

Then  we  have: 

B'V  _  CV 
'AU~'CA"> 

A'V        CV 


^^^  B'U  ~  CB'  ' 
Thus 

A'V     B'V  CV2  CV2 


A'U    B'U        CA'.CB'         CM 2 

Therefore 

A'V  I  M'V        M'V  I  B'V 


/M'V\  2 


A'U    M'U        M'U\  B'U 


Ideal  Reflection. 


257 


Hence  the  Ideal  Length  of  AM  is  equal  to  the  Ideal 
Length  of  MB. 

Thus  we  have  the  following  result: 

Inversion  with  regard  to  a  circle  of  the  system  changes 
any  Ideal  Point  A  ifito  an  Ideal  Point  B,  such  that  the  Ideal 
Line  AB  is  perpendicular  to,  and  ''bisected'  by,  the  Ideal  LÌ7ie 
coinciding  with  the  circle  of  inversion. 


Fig.  95 

In  other  words,  inversion  with  regard  to  such  a  circle 
causes  any  Ideal  Point  A  to  take  the  position  of  its  image  in 
the  corresponding  Ideal  Line. 

We  proceed  to  examine  what  effect  such  inversion  has 
upon  an  Ideal  Line. 

Since  a  circle^   orthogonal  to  the  fundamental  circle, 

17 


2  e  8     Appendix  V.     Impossibility  of  proving  Euclid's  Postulate. 

inverts  into  a  circle  also  orthogonal  to  the  fundamental  circle, 
any  Ideal  Line  AB  inverts  into  another  Ideal  Line  ab,  pass- 
ing through  the  point  M,  where  AB  meets  the  circle  of  in- 
version (cf.  Fig.  96).    Also  the  points  U,  V  invert  into  the 


Fig.  96. 

points  ti  and  v  on  the  fundamental  circle;  and  the  lines  AB 
and  ab  are  equally  inclined  to  the  circle  of  inversion. 

It  is  easy  to  show  that  the  Ideal  Lengths  of  AM  and 
BM  are  equal  to  those  of  aM  a.nd  ^ J/ respectively,  and  it 
follows  that  the  Ideal  Length  of  the  segment  AB  is  unaltered 
by  this  inversion.  Also  we  have  seen  that  Aa  and  Bb  are 
perpendicular  to,  and  'bisected'  by,  the  Ideal  Line  coinciding 
vnth  this  circle. 

//  follows  from  these  results  that  inversion  with  regard 
to  any  circle  of  the  system  has  the  same  effect  upon  an  Ideal 
Segment  as  reflection  in  the  corresponding  Ideal  Line. 

We  are  thus  agaifi  able  to  defi7ie  Ideal  Reflection  in  any 
Ideal  Line  as  the  inversion  of  the  system  of  Ideal  Points  and 


Ideal  Displacement. 


259 


Lines  7mt/i  regard  to  the  circle  which  ^ciacides  with  this 
Ideal  Line. 

It  is  unnecessary  to  define  Ideal  Displacements.,  as  any 
displacement  can  be  obtained  by  a  series  of  reflections  and 
any  Ideal  Displacement  by  a  series  of  Ideal  Reflections. 

We  notice  that  the  definition  of  the  Ideal  Length  of 
any  Segment  fixes  the  Ideal  Unit  of  Length.  We  may  take 
this  on  one  of  the  diameters  of  the  fundamental  circle,  since 
these  lines  are  also  Ideal  Lines  of  the  system.  Let  it  be  the 
segment  OP  (Fig.  97). 


Fig.  97- 

Then 

we 

must  have 

/0V\  PV\ 

l°g  \ou\pu) 

i.  e. 

1     PU 

log    py    =    I. 

Therefore 

PU 

PF  ~  ^' 

and  the  point  P  divides  the  diameter  in  the  ratio  e:  1. 

The  Unit  Segment  is  thus  fixed  for  any  position  in  the 


17=" 


200     Appendix  V.     Impossibility  of  proving  Euclid's  Postulate. 

domain  of  the  Ideal  Points,  since  the  segment  OP  can  be 
'moved'  so  that  one  of  its  ends  coincides  with  any  given 
Ideal  Point. 

A  different  expression  for  the  Ideal  Length 

would  simply  mean  an  alteration  in  the  unit,  and  taking 
logarithms  to  any  other  base  than  <?  would  have  the  same 
effect. 

§  ir.  Some  further  Theorems  in  this  Non-Euclidean 
Geometry. 

We  are  now  in  a  position  to  establish  some  further 
theorems  of  the  Hyperbolic  Geometry  using  the  metrical 
properties  of  this  Ideal  Geometry. 

In  the  first  place  we  can  state  that  Similar  Triangles 
are  impossible  in  this  geometry. 

We  also  see  that  Parallel  Ideal  Lines  are  asymptotic; 
that  is,  these  lines  continually  approach  each  other  and  the 
distance  between  them  tends  to  zero. 

Further,  it  is  obvious  that  as  the  point  A  moves  away 
along  the  perpendicular  MA  to  the  line  BC  (cf.  Fig.  92),  the 
angle  of  parallelism  dimmishes  from  —  to  zero  m  the  limit. 

Again,  we  can  prove  from  the  Ideal  Geometry  that  the 
Angle  of  Parallelism  TT  (/),  corresponding  to  a  segment  /,  is 
given  by 

tan    n  (/)  _  -P 
2 

Consider  an  Ideal  Line  and  the  Ideal  Parallel  to  it 
through  a  point  A. 

Let  AM  (Fig.  98)  be  the  perpendicular  to  the  given  line 
MU^  and  A  U  the  parallel. 

Let  the  figure  be  inverted  from  the  point  J/",  the  radius  of 
inversion  being  the  tangent  from  M"  to  the  fundamental  circle. 


Further  Theorems  in  this  Geometry. 


261 


Then  we  obtain  a  new  figure  (cf.  Fig.  99)  in  which  the 
corresponding  Ideal  Lengths  are  the  same,  since  the  circle 
of  inversion  is  a  circle  of  the  system.  The  lines  AM  and 
MU  become  straight  lines  through  the  centre  of  the  fund- 


amental circle,  which  is  the  inverse  of  the  point  M'. 
Also  the  circle  A  U  becomes  the  circle  a'u^  touching  the 
radius  mu  at  ?/,  and  cutting  via  at  an  angle  TT (/).  These 
radii,  mu,  nib,  are  also  Ideal  Lines  of  the  system. 

The  Ideal  Length  of  the  Segment  AM  is  taken  as  p. 


Then 


(A'B  \M'B\ 

^=^^^^  [ax  lire) 

.         /a'b     I  vi'b  \ 

But  ac  =  k  —  k  tan  ( ) 

(i-"f). 


and  db  =  /&  +  /&  tan 


202     Appendix  V.     Impossibility  of  proving  Euclid's  Postulate. 

where  k  is  the  radius  of  the  fundamental  circle. 

TT  {p^ 
Thus  p  =  log  cot  — ^  ; 


and  e 


-p 


tan 


n(/) 


Fig.  99. 

Finally,  in  this  geometry  there  will  be  three  kinds  of 
circles.  There  will  be  the  circle^  with  its  centre  at  a  finite 
distance;  the  Limiting  Curve  or  Horocycle,  with  its  centre  at 
infinity,  (at  a  point  where  two  parallels  meet)  ;  and  the  Equi- 
distant Curve,  with  its  centre  at  the  imaginary  point  of  inter- 
section of  two  lines  with  a  common  perpendicular. 

The  first  of  these  curves  would  be  traced  out  in  the 
Ideal  Geometry  by  one  end  of  an  Ideal  Segment,  when  it  is 
reflected  in  the  lines  passing  through  the  other  end;  that  is, 
by  the  rotation  of  this  Ideal  Segment  about  that  end.  The 
second  occurs  when  the  Ideal  Segment  is  reflected  in  the 
successive  lines  of  the  pencil  of  Ideal  Lines  all  parallel  to  it 
in  the  same  direction;   and  the  third,   when  the  reflection 


Application  to  Euclid's  Parallel  Postulate.  263 

takes  place  in  the  system  of  Ideal  Lines  which  all  have  a 
perpendicular  with  this  segment.  That  these  correspond  to 
the  common  Circle^  the  Horocycle  and  the  Equidistant  Curve 
of  the  Hyperbolic  Geometry  is  easily  proved. 

§  12.  The  Impossibility  of  Proving  Euclid! s  Parallel 
Postulate. 

We  could  obtain  other  results  of  the  Hyperbolic  Geo- 
metry, and  find  some  of  its  constructions,  by  further  examin- 
ation of  the  properties  of  this  set  of  circles;  but  this  is  not 
our  object.  Our  argument  was  directed  to  proving,  by  reas- 
oning involving  only  elementary  geometry,  that  it  is  impossible 
for  any  inconsistency  or  contradiction  to  arise  in  this  Non- 
Euclidean  Geometry.  If  such  contradiction  entered  into  this 
Plane  Geometry,  it  would  also  occur  in  the  interpretation  of 
the  result  in  the  Ideal  Geometry.  Thus  the  contradiction 
would  also  be  found  in  the  Euclidean  Geometry.  We  can, 
therefore,  state  that  it  is  impossible  that  any  logical  incon- 
sistency could  be  traced  in  the  Plane  Hyperbolic  Geometry.  It 
could  still  be  argued  that  such  contradiction  might  be  found 
in  the  Solid  Hyperbolic  Geometry.  An  answer  to  this  ob- 
jection is  at  once  forthcoming.  The  geometry  of  the  system 
of  circles,  all  orthogonal  to  a  fixed  circle,  can  be  at  once 
extended  into  a  three  dimensional  system.  The  Ideal  Points 
are  taken  as  th£  pairs  of  points  inverse  to  a  fixed  sphere, 
excluding  the  points  on  the  surface  of  the  sphere  from  their 
domain.  The  Ideal  Lines  are  the  circles  tlurough  two  Ideal 
Points.  The  Ideal  Planes  are  the  spheres  through  three  Ideal 
Points,  not  lying  on  an  Ideal  Line.  The  ordinary  plane  enters 
as  a  particular  case  of  these  Ideal  Planes,  and  so  the  Plane 
Geometry  just  discussed  is  a  special  case  of  a  plane  geo- 
metry on  this  system.  With  suitable  definitions  of  Ideal 
Lengths,  Ideal  Parallels  and  Ideal  Displacements,  we  have 
a  Solid  Geometry  exactly  analogous  to  the  Hyperbolic  Solid 


204     Appendix  V.     Impossibility  of  proving  Euclid's  Postulate. 

Geometry.  It  follows  that  no  logical  inconsistency  can  exist 
in  the  Hyperbolic  Solid  Geometry,  since  if  there  were  such 
a  contradiction,  it  would  also  be  found  in  the  interpretation 
of  the  result  in  this  Ideal  Geometry;  and  therefore  it  would 
enter  into  the  Euclidean  Geometry. 

By  this  result  our  argument  is  complete.  However  far 
the  HyperboUc  Geometry  were  developed,  no  contradictory 
results  could  be  obtained.  This  system  is  thus  logically 
possible;  and  the  axioms  upon  which  it  is  founded  are  not 
contradictory.  Hence  it  is  impossible  to  prove  Euclid's 
Parallel  Postulate,  since  its  proof  would  involve  the  denial 
of  the  Parallel  Postulate  of  Bolvai-Lobatschewsky. 


Index  of  Authors. 


[The  Jtnmbers  refer  to  pages.] 


Aganis,  (6th  Century^  8— ii. 

Al-Nirizi,  (9th  Century).  7,  9. 

Andrade,  J.  181,   194. 

Archimedes,  (287—212).  9,  Tl 
23,  25,  30,  34,  37,  46,  56,  59 
119 — 121,  144,  181,  183. 

Aristotle,  (384—322).  4,  8,  18,  19 

Arnauld,   (1612— 1694).   17. 

Baltzer,R.  (1818  — 1887).  121—3 

Barozzi,  F.  (l6'h  Century).  12. 

Battels,  J.  M.  C.  (1769—1836) 
84,  91—2. 

Battaglini,  G.  (1826—1894).  86 
100,  122,  126 — 7. 

Beltrami,  E.  (1835—1900).  44 
122,  126—7,  ^33.  13S— 9.  145 
147,  T[73— 5.  234—6. 

Bernoulli,  D.  (1700—1782).  192. 

Bernoulli,  J.  (1744—1807).  44. 

Bessel,  F.  W.  (1784-1846).  65 
-67. 

Besthorn,  R.  O.  7. 

Bianchi,  L.  129,  135,  209. 

Biot,  J.  B.  (1774—1862).  52. 

Boccardini,  G.  44. 

Bolyai,  J.  (1802 — 1860).  51,  61, 
65,  74,  96—107,  109-116- 
121—6, 128, 137, 141, 145, 147, 
152,  154,  157— 8,  161,  164, 
170,  173—5.  177—8,  193—4, 
200,  222,  225,  233,  238,  249, 
252—3,  264. 


Bolyai,  W.  (1775  —  1856).  55»  60 

— 1,   65 — 6,    74,    96.  98—101, 

120,  125 — 6. 
Boncampagni,   B.    (1821  —1894). 

125. 
Bonola,  R.  (1875—1611)  15,  26, 

30,  115,  176—7,  220. 
Borelli,  G.  A.  (1608—1679).   11, 

13,  17- 
Boy,  W.  149. 

Campanus,  G.  (13'!^  Century).  17. 
Candalla,  F.  (1502—1594).  17. 
Carnet,  L.  N.M.  (1753—1823).  53- 
Carslaw,  H.  S.  40,  238. 
Cassani,  P.  (1832  — 1905).   127. 
Castillon,  G,  (1708  —  1791).  12. 
Cataldi,  P.  A.  (1548?— 1626).  13. 
Cauchy,  A.  L.  (1789—1857).  199. 
Cayley,    A.    (1821  —  1895).    127, 

148,  156,  163—4,  174,  179. 
Chasles,  M.  (1793—1880).  155. 
Clavio,  C.  (1537—1612).  13,  17. 
Clebsch,  A.  (1833—1872).  161. 
Clifford.W.K.  (1845—1879).  139, 

142,  200—215. 
Codazzi,  D.  (1824—1873).  137. 
Commandino,    F.    (1509—1575). 

12,  17. 
Coolidge,  J.  L.   129. 
Couturat,  L.  54. 
Cremona,  L.   (1830—1903).  123, 

127. 


266 


Index  of  Authors. 


Curtze,  M.  (1837—1903).  7. 
D'Alembert,    J.    le    R.     (1717— 

1783)-  52'  54,  192,  197—8. 
Dedekind,  J.W.R.  (1831-1899). 

139- 
Dehn,  M.  30,  120,  144. 
Delambre,  J.  B.  J.  (1749 — 1822). 

198. 
De  Morgan,  A.  (1806—1870).  52. 
Dickstein,  S.  139. 
Duhem,  P.  182. 
Eckwehr,  J.  W.  v.  (1789-1857). 

99- 

Engel,  F.  16,  44—5,  50,  60,  64, 
66,  83—6,  88,  92—3,  96,  101, 
216 — 7,  220. 

Enriques,  F.  156,  166,  183,  225. 

Eòtvòs,  125. 

Euclid  (circa  330—275).  1  —  8,  10, 
12 — 14,  16—20,  22,  38,  51 — 2, 
54—5,  61—2,  68,  75,  82,  85, 
92,  95,  loi— 2,  104,  110,  112, 
118 — 120,  127,  139,  141,  147, 
152,  154—5,  ^57,  164,  176— 
i8i,  183,  191 — 5,  199—201, 
227,  237—9,  241,  267. 

Fano,  G.  153. 

Flauti,  V.  (1782— 1863)  12. 

Fleischer,  H.  156. 

Flye  St.  Marie,  91. 

Foncenex,  D.  de,  (1743 — 1799). 
53,   146,  190—2,   197—8. 

Forti,  A.  (1818 — ).   122,  124—5. 

Fourier,J.B.  (1768— 1830).  54 — 5. 

Frattini,  G.   127. 

Frankland,  W.  B.  2,  63. 

Friedlein,  G.  2. 

Frischauf,  J.  100,  126. 

Gauss,  C.  F.  (1777—1855).  16, 
60—68,  70—78,  83—4,  86,  88, 
90 — 2,  99 — 101,  110,  113,  122 


—3,   127,  131,  135,  152,  177, 

200. 
Geminus,  (ist.  Century,  B.  C).  3, 

7,  20. 
Genocchi,  A.  (1817 — 18S9).  145, 

191,  198—9. 
Gerling,    Ch.    L.     (1788—1864). 

65,  66,  76 — 7,  121 — 2. 
Gherardo  daCremona,  (12th  Cen- 
tury). 7. 
Giordano  Vitale,  (1633 — 1711).  I4 

—  15,  17,  26. 
Goursat,  E.  145. 
I    Gregory,  D.  (1661  — 1710)  17,  20. 
I    Grossman,  M.   169,  225. 

Giinther,  S.  127. 
i    Halsted,  G.  B.  44,  86, 93,  100,  139. 
Hauff,  J.  K.  F.  (1766—1846).  75. 
Heath,  T.  L.  1,  2,  63. 
Heiberg,  J.  L.  1,  2,  7,  181. 
Heilbronner,  J.  C.  (1706 — 1745). 

44. 
Helmholtz,   H.   v.   (1821—1894). 

126,  145,  152—3,  176—7,  179. 
Hilbert,  D.  23,  145 — 6,  222 — 4, 

236. 
Hindenburg,  K.  F.  (1741 — 1808). 

45- 
Hoffman,  J.  (1777—1866).   12. 
Holmgren,  E.  A.  145 — 6. 
Hoiiel,  J.  (1823—1866).  52,   86, 

100,  121,  123—7, 139,^47, 152- 
Kant,  I.  (1724—1804).  64,  92,121. 
Kastner,  A.  G.  (1719—1800).  50, 

60,  64,  66. 
Killing,  W.  91,  215. 
Klein,  K.  F.  129,  138,  148,  153, 

158,    164,   17Ó,   180,  200,  211, 

213—5,  236—7. 
Kliigel,  G.  S.  (1739— 1812).   12, 

44,  51,  64,  77,  92. 


Index  of  Authors. 


267 


Kiirschàk,  J.  113. 

Lagrange,  J.  L.  (1736— 1813).  53 

—4,  182—3,  198. 
Laguerre,    E.    N.    (1S34— 1866). 

155- 
Lambert,  J.  H.  (1728—1777).  44 
—51,    58,    65—6,    74,   77-8, 
81—2,   92,  97,  107,  129,  139, 

144- 
Laplace,  P.  S.  (1749  —  1827),    53 

-54,  198- 
Legendre,   A.    M.    (1752  — 1833Ì, 

29.  44,  55—59.  74>  §4,  88,  92, 
122,  128,  139,  144- 
Leibnitz,  G.  W.  F.  (1646—1716). 

54- 

Lie,  S.  (1842 — 1899).  152—4,  179. 

Liebmann,  H.   86,  89,   113,  145, 
180,  220. 

Lindemann,  F.  161. 

Lobatschewsky,  N.  J.  (1793  — 
1856).  44,  5L  55>  63,  65,  74 
80,  84 — 99,  101—3,  104—6; 
111—3,  116,  121 — 8,  137, 141 
145,  H7,  152,  154,  157—8 
161,  164,  170,  173—5,  177—8 
193 — 4,  217,  220,  222,  225 
238,  249,  252—3,  264. 

Lorenz,  J.  F.  (1738—1807).  58, 
120. 

Lukat,  M.  129. 

Liitkemeyer,  G.  145 — 6. 

Mc  Cormack,  T.  J.  182. 

Mach,  E.  181. 

Minding,  F.  (1806—1885).    132, 

^37- 
Mobius,  A.  F.  (1790—1868).  148 

— 9- 

Monge,  G.  (1746—1818).   54—5. 
Montucla,    J.    E.     (1725  — 1799) 
44.  92. 


Nasìr-Eddìn,  (1201  — 1274).  10, 
12—3,   16,  37—8,  120. 

Newton,  L  (1642—1727).  53. 

01bers,H.W.M.(i758— 1840).  65. 

Oliver,  (ist  Half  of  the  17*  Cent- 
ury). 17. 
j    Ovidio,  (d')  E.  127. 

Paciolo,  Luca  (circa  1445  — 1514)- 

17- 

Pascal,  E.   127,  139. 

Pasch,  M.   176. 

Picard,  C.  É.  128. 

Poincaré,  H.  154,  180. 

Poncelet,  J.  V.  (1788—1867).  155, 
236. 

Posidonius,  (ist  Century  B.C.I  2, 
3,  8,  14. 

Proclus,  (410— 485^  2—7,  12—3, 
18—20,   119. 

Ptolemy,  (87—165).  3—4,  119. 

Riccardi,  P.  (182S— 1898).  17. 

Ricordi,  E.  127. 

Riemann,  B.  (1826—1866).  126, 
129,  138—9,  141—3,145  —  154, 
157—8,  160—1,  163—4,  175 
—  7,  179—180,  194,  201—2. 

Saccheri,  G.  (1667—1733).  4, 
22—4,  26,  28—30,  34,  36—46, 
51,  55-7,  65—6,  78,  85,  87 
—8,  97,    120,   129,   139,  141, 

144- 
Sartorius   v.  Waltershausen,   W. 

(1809—1876).   122. 
Saville,  H.  (1549—1622).  17. 
Schmidt,   F.   (1826 — 1901).    121, 

124—5. 
Schumacher,  H.  K.  (1780—1850). 

65-7,  75,  122—3,  152. 
Schur,  F.  H.   176. 
Schweikart,  F.  K.  (1780—1859). 

67,  75  -78,  80,  83,86, 107, 122. 


268 


Index  of  Authors. 


Segre,  C.  44,  66,  77— S,  92. 

Seyffer,  K.  F.  (1762— 1S22).  60, 66. 

Simon,  H.  91. 

Simplicius,  (6'h  Century).    8,  10. 

Sintsoff,  D.  139. 

Stackel,  P.  16,  44—5,  50,  60  —  1, 

63,    66,    82 — 3,    lOT,    112—3, 

124-5. 
Staudt.G.  C.v.  (1798—1867).  129, 

154,  233,  236. 
Szasz,  C.  (1798—1853).  97. 
Tannery,  P.  (1S43— 1904).  7,  20. 
Tacquet,  A.  (1612  —  1660).  17. 
Tartaglia,  N.  (1500—1557).  17. 
Taurinus,    F.    A.    (1794—1874). 

65—6,  74,  77-9'  Si— 3,  87, 

89-91,  94,  99,  112,  137,  173. 


Thibaut,  B.  F.  (1775— 1S32).  63. 

Tilly    (de),    J.  M.    55,  114,  194. 

Townsend,  E.  J.  236. 

Vailati,  G.  18,  22. 

Valerio  Luca  (?  1522 — 1618).  17. 

Vasiliev,  A.  93, 

Wachter,  F.  L.  (1792—1817).  62 

—3,  66,  88. 
Wallis,  J.    (1616 — 1703).  12,  15 

—7,  29,  53'  120. 
Weber,  H.  180. 
Wellstein,  J.  180,  23S. 
Zamberti,   B.    (ist    Half    of    the 

l6th  Century).  17. 
Zeno,  (495— 435V  6. 
Zolt,  A.  (de)  127. 


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£ssays  on  the  Theory  of  Numbers, 

(1)  Continuity  and  Irrational  Numbers,  (2)  The  Nature 
and  ]\Ieaning  of  Numbers.  By  Richard  Dedekind.  From 
the  German  by  W.  W.  Beman.  Pages,  115.  Cloth,  75 
cents  net.     (3s.  6d.  net.) 

These  essays  mark  one  of  the  distinct  stages  in  the  devel- 
opment of  the  theory  of  numbers.  They  give  the  founda- 
tion upon  which  the  whole  science  of  numbers  may  be  es- 
tablished. The  first  can  be  read  without  any  technical, 
philosophical  or  mathematical  knowledge  ;  the  second  re- 
quires more  power  of  abstraction  for  its  perusal,  but  power 
of  a  logical  nature  only. 

"A  model  of  clear  and  beautiful  reasoning." 

— Journal  of  Physical  Chemistry. 

"The  work  of  Dedekind  is  very  fundamental,  and  I  am  glad  to  have  it 
in  thi.=!  carefully  wrought  English  version.  I  think  the  book  should  be 
of   much    service   to   American    mathematicians    and    teachers." 

— Prof.  E.  H.  Moore,  University  of  Chicago. 

"It  is  to  be  hoped  that  the  translation  will  make  the  essays  better 
known  to  English  mathematicians  ;  they  are  of  the  very  first  importance, 
and  rank  with  the  work  of  Weierstrass,  Kronecker,  and  Cantor  in  the 
same  field." — Nature, 


Elementary  Illustrations  of  the  Differential 
and  Integral  Calculus. 

By  Augustus  De  Morgan.  New  reprint  edition.  With 
subheadings  and  bibliography  of  English  and  foreign  works 
on  the  Calculus.     Price,  cloth,  $1.00  net.     (4s.  6d  net.) 

"It  aims  not  at  helping  students  to  cram  for  examinations,  but  to  give 
a  scientific  explanation  of  the  rationale  of  these  branches  of  mathe- 
matics. Like  all  that  De  Morgan  wrote,  it  is  accurate,  clear  and 
philosophic." — Literary   World,  London. 


On   the    Study   and    Difficulties   of   Mathe- 
matics. 

By  Augustus  De  Morgan.  With  portrait  of  De  Morgan, 
Inde.x,  and  Bibliographies  of  Modern  Works  on  Algebra, 
the  Philosophy  of  Mathematics,  Pangeometry,  etc.  Pages, 
viii,  288.     Cloth,  $1.25  net.     (5s.  net.) 

"The  point  of  view  is  unusual  ;  we  are  confronted  by  a  genius,  who, 
like  his  kind,  shows  little  heed  for  customary  conventions.  The  'shak- 
ing up'  which  this  little  work  will  give  to  the  young  teacher,  the  stim- 
ulus and  implied  criticism  it  can  furnish  to  the  more  experienced,  make» 
its  possession   most  desirable." — Michigan  Alumnus. 


The  Open  Court  Mathematical  Series 


The  Foundations  of  Geometry. 

By  David  Hilbert,  Ph.  D.,  Professor  of  Mathematics  in 
the  University  of  Gottingen.  With  many  new  additions 
still  unpublished  in  German.  Translated  by  E.  J.  Town- 
send,  Ph.  D.,  Associate  Professor  of  Mathematics  in  the 
University  of  Illinois.  Pages,  viii,  133.  Cloth,  $1.00  net. 
(4s.  6d  net.) 

"Professor  Hilbert  has  become  so  well  known  to  the  mathematical 
world  by  his  writings  that  the  treatment  of  any  topic  by  him  commands 
the  attention  of  mathematicians  everywhere.  The  teachers  of  elemen- 
tary geometry  in  this  country  are  to  be  congratulated  that  it  is  possible 
for  them  to  obtain  in  English  such  an  Important  discussion  of  these 
points   by   such  an   authority." — Journal   of  Pedagogy. 

Euclid's  Parallel  Postulate  :  Its  Nature, Val- 
idity and  Place  in  Geometrical  Systems. 

By  John  William  Withers,  Ph.  D.  Pages  vii,  192.  Cloth, 
net  $1.25.     (4s.  6d.  net.) 

"This  is  a  philosophical  thesis,  by  a  writer  who  is  really  familiar  with 
the  subject  on  non-Euclidean  geometry,  and  as  such  it  is  well  worth 
reading.  The  first  three  chapters  are  historical  ;  the  remaining  three 
deal  with  the  psychological  and  metaphysical  aspects  of  the  problem  ; 
finally  there  is  a  bibliography  of  fifteen  pages.  Mr.  Withers's  critique, 
on  the  whole,  is  quite  sound,  although  there  are  a  few  passages  either 
vague  or  disputable.  Mr.  Withers's  main  contention  is  that  Euclid's 
parallel  postulate  is  empirical,  and  this  may  be  admitted  in  the  sense 
that  his  argument  requires  ;  at  any  rate,  he  shows  the  absurdity  of 
some  statements  of  the  a  priori  school." — Nature. 

Mathematical  Essays  and  Recreations* 

By   Hermann    Schubert,    Professor   of   Mathematics    in 

Hamburg.     Contents:     Notion  and  Definition  of  Number; 

Monism  in  Arithmetic  ;    On  the   Nature  of   Mathematical 

Knowledge  ;  The  Magic  Square  ;  The  Fourth  Dimension  ; 

The  Squaring  of  the  Circle.     From  the  German  by  T.  J. 

McCormack.     Pages,  149.     Cuts,  37.     Cloth,  75  cents  net. 

(3s.  6d.  net.) 
"Professor  Schubert's  essays  make  delightful  as  well  as  instructive 
reading.  They  deal,  not  with  the  dry  side  of  mathematics,  but  with  the 
philosophical  side  of  that  science  on  the  one  hand  and  its  romantic  and 
mystical  side  on  the  other.  No  great  amount  of  mathematical  knowl- 
edge is  necessary  in  order  to  thoroughly  appreciate  and  enjoy  them. 
They  are  admirably  lucid  and  simple  and  answer  questions  in  which 
every  intelligent  man  is  interested." — Chicac/o  Evening  Post. 
"They  should  delight  the  jaded  teacher  of  elementary  arithmetic,  who 
is  too  liable  to  drop  into  a  mere  rule  of  thumb  system  and  forget  the 
scientific  side  of  his  work.  Their  chief  merit  is  however  their  intel- 
ligibility. Even  the  lay  mind  can  understand  and  take  a  deep  interest 
in  what  the  German  professor  has  to  say  on  the  history  of  magic 
squares,  the  fourth  dimension  and  squaring  of  the  circle." 

— Saturday  Review. 


The  Open  Court  Mathematical  Series 


Geometric  Exercises  in  Paper-Foldiné» 

By  T.  SuNDARA  Row.  Edited  and  revised  by  W.  W.  Be- 
MAN  and  D.  E.  Smith.  With  half-tone  engravings  from 
photographs  of  actual  exercises,  and  a  package  of  papers 
for  folding.  Pages,  x,  148.  Price,  cloth,  $1.00  net.  (43. 
6d.  net.) 

"The  book  i.s  simply  a  revelation  in  paper  folding.  All  sorts  of  things 
are  done  with  the  paper  squares,  and  a  large  number  of  geometric 
figures  are  constructed   and   explained  in  the  simplest  way." 

— Teachers'  Itistitute. 

Maéic  Squares  and  Cubes. 

By  W.  S.  Andrews.  With  chapters  by  Paul  Carus,  L.  S. 
Frierson  and  C.  A.  Browne,  Jr.,  and  Introduction  by 
•Paul  Carus.  Price,  $1.50  net.  (7s.6d.net.) 
The  first  two  chapters  consist  of  a  general  discussion  of  the 
general  qualities  and  characteristics  of  odd  and  even  magic 
squares  and  cubes,  and  notes  on  their  construction.  The 
third  describes  the  squares  of  Benjamin  Franklin  and  their 
characteristics,  while  Dr.  Carus  adds  a  further  analysis 
of  these  squares.  The  fourth  chapter  contains  "Reflections 
on  Magic  Squares"  by  Dr.  Carus,  in  which  he  brings  out 
the  intrinsic  harmony  and  symmetry  which  exists  in  the 
laws  governing  the  construction  of  these  apparently  mag- 
ical groups  of  numbers.  Mr.  Frierson's  "Mathematical 
Study  of  Magic  Squares,"  which  forms  the  fifth  chapter, 
•  states  the  laws  in  algebraic  formulas.  Mr.  Browne  con- 
tributes a  chapter  on  "Magic  Squares  and  Pythagorean 
Numbers,"  in  which  he  shows  the  importance  laid  by  the 
ancients  on  strange  and  mystical  combinations  of  figures. 
The  book  closes  with  three  chapters  of  generalizations  in 
which  Mr.  Andrews  discusses  "Some  Curious  Magic 
Squares  and  Combinations,"  "Notes  on  Various  Con- 
structive Plans  by  Which  Magic  Squares  May  Be  Classi- 
fied," and  "The  Mathematical  Value  of  Alagic  Squares." 

"The  examples  are  numerou.s  ;  the  laws  and  rules,  some  of  them 
original,  for  making  squares  are  well  worked  out.  The  volume  is 
attractive  in  appearance,  and  what  is  of  the  greatest  importance  in 
such    a    work,    the    proof-reading    has    been    careful." — The    Nation. 

The  Foundations  of  Mathematics. 

A  Contribution  to  The  Philosophy  of  Geometry.  By  Dr. 
Paul  Carus.  140  pages.  Cloth.  Gilt  top.  75  cents  net. 
(3s.   6d.   net.) 


The  Open  Court  Publishing  Co. 

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